# Copyright 2011, 2012, 2013, 2014 Kevin Ryde
# This file is part of Math-PlanePath.
#
# Math-PlanePath is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by the
# Free Software Foundation; either version 3, or (at your option) any later
# version.
#
# Math-PlanePath is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
# or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
# for more details.
#
# You should have received a copy of the GNU General Public License along
# with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
# Maybe:
# "Turn" N of turn positions
# $path->n_next_turn($n)
package Math::NumSeq::PlanePathN;
use 5.004;
use strict;
use Carp;
use constant 1.02;
use vars '$VERSION','@ISA';
$VERSION = 115;
use Math::NumSeq;
@ISA = ('Math::NumSeq');
use Math::NumSeq::PlanePathCoord;
# uncomment this to run the ### lines
# use Smart::Comments;
sub description {
my ($self) = @_;
if (ref $self) {
return "N values on $self->{'line_type'} of path $self->{'planepath'}";
} else {
# class method
return 'N values from a PlanePath';
}
}
use constant::defer parameter_info_array =>
sub {
return [
Math::NumSeq::PlanePathCoord::_parameter_info_planepath(),
{ name => 'line_type',
display => 'Line Type',
type => 'enum',
default => 'X_axis',
choices => ['X_axis',
'Y_axis',
'X_neg',
'Y_neg',
'Diagonal',
'Diagonal_NW',
'Diagonal_SW',
'Diagonal_SE',
'Depth_start',
'Depth_end',
],
description => 'The axis or line to take path N values from.',
},
];
};
#------------------------------------------------------------------------------
my %oeis_anum =
(
# MultipleRings,step=0 -- integers 1,2,3, etc, but starting i=0
);
sub oeis_anum {
my ($self) = @_;
### PlanePathN oeis_anum() ...
my $planepath_object = $self->{'planepath_object'};
{
my $key = Math::NumSeq::PlanePathCoord::_planepath_oeis_anum_key($self->{'planepath_object'});
my $i_start = $self->i_start;
if ($i_start != $self->default_i_start) {
### $i_start
### cf n_start: $planepath_object->n_start
$key .= ",i_start=$i_start";
}
### planepath: ref $planepath_object
### $key
### whole table: $planepath_object->_NumSeq_N_oeis_anum
### key href: $planepath_object->_NumSeq_N_oeis_anum->{$key}
if (my $anum = $planepath_object->_NumSeq_N_oeis_anum->{$key}->{$self->{'line_type'}}) {
return $anum;
}
if (my $anum = $planepath_object->_NumSeq_N_oeis_all_anum->{$self->{'line_type'}}) {
return $anum;
}
}
{
my $key = Math::NumSeq::PlanePathCoord::_planepath_oeis_key($planepath_object);
my $i_start = $self->i_start;
if ($i_start != $self->default_i_start) {
### $i_start
### cf n_start: $planepath_object->n_start
$key .= ",i_start=$i_start";
}
### $key
### hash: $oeis_anum{$key}
return $oeis_anum{$key}->{$self->{'line_type'}};
}
}
#------------------------------------------------------------------------------
sub default_i_start {
my ($self) = @_;
my $planepath_object = $self->{'planepath_object'}
# nasty hack allow no 'planepath_object' when SUPER::new() calls rewind()
|| return 0;
my $method = "_NumSeq_$self->{'line_type'}_i_start";
if (my $func = $planepath_object->can($method)) {
return $planepath_object->$func();
}
return 0;
}
sub new {
my $self = shift->SUPER::new(@_);
my $planepath_object = ($self->{'planepath_object'}
||= Math::NumSeq::PlanePathCoord::_planepath_name_to_object($self->{'planepath'}));
### $planepath_object
my $line_type = $self->{'line_type'};
### i_func name: "i_func_$line_type"
$self->{'i_func'}
= $self->can("i_func_$line_type")
|| croak "Unrecognised line_type: ",$line_type;
$self->{'pred_func'}
= $self->can("pred_func_$line_type")
|| croak "Unrecognised line_type: ",$line_type;
if (my $func
= $planepath_object->can("_NumSeq_${line_type}_step")) {
$self->{'i_step'} = $planepath_object->$func();
} elsif ($planepath_object->_NumSeq_A2()
&& ($line_type eq 'X_axis'
|| $line_type eq 'Y_axis'
|| $line_type eq 'X_neg'
|| $line_type eq 'Y_neg')) {
$self->{'i_step'} = 2;
} else {
$self->{'i_step'} = 1;
}
### i_step: $self->{'i_step'}
# for use in pred()
$self->{'i_start'} = $self->i_start;
$self->rewind;
return $self;
}
sub rewind {
my ($self) = @_;
$self->{'i'} = $self->i_start;
}
sub next {
my ($self) = @_;
### NumSeq-PlanePathN next(): "i=$self->{'i'}"
my $i = $self->{'i'};
my $n = &{$self->{'i_func'}} ($self, $i);
### $n
if (! defined $n) {
### i_func returns undef, no value ...
return;
}
# secret experimental automatic bigint to preserve precision
if (! ref $n && $n > 0xFF_FFFF) {
$n = &{$self->{'i_func'}}($self,_to_bigint($i));
}
return ($self->{'i'}++, $n);
}
sub _to_bigint {
my ($n) = @_;
# stringize to avoid UV->BigInt bug in Math::BigInt::GMP version 1.37
return _bigint()->new("$n");
}
# or maybe check for new enough for uv->mpz fix
use constant::defer _bigint => sub {
# Crib note: don't change the back-end if already loaded
unless (Math::BigInt->can('new')) {
require Math::BigInt;
eval { Math::BigInt->import (try => 'GMP') };
}
return 'Math::BigInt';
};
sub ith {
my ($self, $i) = @_;
### NumSeq-PlanePathN ith(): $i
return &{$self->{'i_func'}}($self, $i);
}
sub i_func_X_axis {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n ($i * $self->{'i_step'},
$path_object->_NumSeq_X_axis_at_Y);
}
sub i_func_Y_axis {
my ($self, $i) = @_;
### i_func_Y_axis(): "i=$i"
### X: $self->{'planepath_object'}->_NumSeq_Y_axis_at_X
### Y: $i * $self->{'i_step'}
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n ($path_object->_NumSeq_Y_axis_at_X,
$i * $self->{'i_step'});
}
sub i_func_X_neg {
my ($self, $i) = @_;
### i_func_X_neg(): $i
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n (-$i * $self->{'i_step'},
$path_object->_NumSeq_X_axis_at_Y);
}
sub i_func_Y_neg {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n ($path_object->_NumSeq_Y_axis_at_X,
- $i * $self->{'i_step'});
}
sub i_func_Diagonal {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n ($i + $path_object->_NumSeq_Diagonal_X_offset,
$i);
}
sub i_func_Diagonal_NW {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n (-$i + $path_object->_NumSeq_Diagonal_X_offset,
$i);
}
sub i_func_Diagonal_SW {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n (-$i + $path_object->_NumSeq_Diagonal_X_offset,
-$i);
}
sub i_func_Diagonal_SE {
my ($self, $i) = @_;
my $path_object = $self->{'planepath_object'};
return $path_object->xy_to_n ($i + $path_object->_NumSeq_Diagonal_X_offset,
-$i);
}
sub i_func_Depth_start {
my ($self, $i) = @_;
### i_func_Depth_start(): "i=$i"
return $self->{'planepath_object'}->tree_depth_to_n($i);
}
sub i_func_Depth_end {
my ($self, $i) = @_;
return $self->{'planepath_object'}->tree_depth_to_n_end($i);
}
#------------------------------------------------------------------------------
sub pred {
my ($self, $value) = @_;
### PlanePathN pred(): $value
my $planepath_object = $self->{'planepath_object'};
unless ($value == int($value)) {
return 0;
}
my ($x,$y) = $planepath_object->n_to_xy($value)
or return 0;
return &{$self->{'pred_func'}} ($self, $x,$y, $value);
}
sub pred_func_X_axis {
my ($self, $x,$y) = @_;
return ($x >= $self->{'i_start'}
&& $y == $self->{'planepath_object'}->_NumSeq_X_axis_at_Y);
}
sub pred_func_Y_axis {
my ($self, $x,$y) = @_;
return ($x == $self->{'planepath_object'}->_NumSeq_Y_axis_at_X
&& $y >= $self->{'i_start'});
}
sub pred_func_X_neg {
my ($self, $x,$y) = @_;
return ($x <= - $self->{'i_start'} && $y == 0);
}
sub pred_func_Y_neg {
my ($self, $x,$y) = @_;
return ($x == 0 && $y <= - $self->{'i_start'});
}
sub pred_func_Diagonal {
my ($self, $x,$y) = @_;
$x -= $self->{'planepath_object'}->_NumSeq_Diagonal_X_offset;
return ($x >= $self->{'i_start'} && $x == $y);
}
sub pred_func_Diagonal_NW {
my ($self, $x,$y) = @_;
return ($x <= - $self->{'i_start'} && $x == -$y);
}
sub pred_func_Diagonal_SW {
my ($self, $x,$y) = @_;
return ($x <= - $self->{'i_start'} && $x == $y);
}
sub pred_func_Diagonal_SE {
my ($self, $x,$y) = @_;
return ($x >= $self->{'i_start'} && $x == -$y);
}
sub pred_func_Depth_start {
my ($self, $x,$y, $n) = @_;
return path_tree_n_is_depth_start($self->{'planepath_object'}, $n);
}
sub pred_func_Depth_end {
my ($self, $x,$y, $n) = @_;
return path_tree_n_is_depth_end($self->{'planepath_object'}, $n);
}
# Return true if $n is the start of a depth level.
sub path_tree_n_is_depth_start {
my ($path, $n) = @_;
my $depth = $path->tree_n_to_depth($n);
return (defined $depth && $n == $path->tree_depth_to_n($depth));
}
# Return true if $n is the end of a depth level.
sub path_tree_n_is_depth_end {
my ($path, $n) = @_;
my $depth = $path->tree_n_to_depth($n);
return (defined $depth && $n == $path->tree_depth_to_n_end($depth));
}
#------------------------------------------------------------------------------
use constant characteristic_integer => 1; # integer Ns
sub characteristic_increasing {
my ($self) = @_;
### PlanePathN characteristic_increasing(): $self
my $method = "_NumSeq_$self->{'line_type'}_increasing";
my $planepath_object = $self->{'planepath_object'};
### planepath_object: ref $planepath_object
### $method
### can code: $planepath_object->can($method)
### result: $planepath_object->can($method) && $planepath_object->$method()
return $planepath_object->can($method) && $planepath_object->$method();
}
sub characteristic_increasing_from_i {
my ($self) = @_;
### PlanePathN characteristic_increasing_from_i(): $self
my $planepath_object = $self->{'planepath_object'};
my $method = "_NumSeq_$self->{'line_type'}_increasing_from_i";
### $method
if ($method = $planepath_object->can($method)) {
### can: $method
return $planepath_object->$method();
}
return ($self->characteristic('increasing')
? $self->i_start
: undef);
}
sub characteristic_non_decreasing {
my ($self) = @_;
### PlanePathN characteristic_non_decreasing() ...
my $planepath_object = $self->{'planepath_object'};
my $method = "_NumSeq_$self->{'line_type'}_non_decreasing";
return (($planepath_object->can($method) && $planepath_object->$method())
|| $self->characteristic_increasing);
}
sub values_min {
my ($self) = @_;
### PlanePathN values_min() ...
my $method = "_NumSeq_$self->{'line_type'}_min";
my $planepath_object = $self->{'planepath_object'};
if (my $coderef = $planepath_object->can($method)) {
### $coderef
return $planepath_object->$coderef();
}
return $self->ith($self->i_start);
}
sub values_max {
my ($self) = @_;
my $method = "_NumSeq_$self->{'line_type'}_max";
my $planepath_object = $self->{'planepath_object'};
if (my $coderef = $planepath_object->can($method)) {
return $planepath_object->$coderef();
}
return undef;
}
{ package Math::PlanePath;
sub _NumSeq_X_axis_i_start {
my ($self) = @_;
my $x_minimum = $self->x_minimum;
if (defined $x_minimum && $x_minimum > 0) {
return int($x_minimum);
}
return 0;
}
sub _NumSeq_Y_axis_i_start {
my ($self) = @_;
### _NumSeq_Y_axis_i_start() ...
my $y_minimum = $self->y_minimum;
if (defined $y_minimum && $y_minimum > 0) {
return int($y_minimum);
}
return 0;
}
sub _NumSeq_Y_axis_at_X {
my ($self) = @_;
my $x_minimum = $self->x_minimum;
if (defined $x_minimum && $x_minimum > 0) {
return $x_minimum;
}
return 0;
}
sub _NumSeq_X_axis_at_Y {
my ($self) = @_;
my $y_minimum = $self->y_minimum;
if (defined $y_minimum && $y_minimum > 0) {
return $y_minimum;
}
return 0;
}
# i_start = Xminimum - Xoffset
# gives Xstart = i_start + Xoffset = Xminimum to start at Xminimum
use List::Util;
sub _NumSeq_Diagonal_i_start {
my ($self) = @_;
my $x_minimum = $self->x_minimum;
my $y_minimum = $self->y_minimum;
return List::Util::max
(int(($x_minimum||0) - $self->_NumSeq_Diagonal_X_offset),
int($y_minimum||0),
0);
}
use constant _NumSeq_Diagonal_X_offset => 0;
use constant _NumSeq_N_oeis_anum => {};
use constant _NumSeq_N_oeis_all_anum => {};
use constant _NumSeq_Depth_start_increasing => 1;
use constant _NumSeq_Depth_end_increasing => 1;
# sub _NumSeq_pred_X_axis {
# my ($path, $value) = @_;
# return ($value == int($value)
# && ($path->x_negative || $value >= 0));
# }
# sub _NumSeq_pred_Y_axis {
# my ($path, $value) = @_;
# return ($value == int($value)
# && ($path->y_negative || $value >= 0));
# }
}
{ package Math::PlanePath::SquareSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
sub _NumSeq_X_neg_increasing {
my ($self) = @_;
return ($self->{'wider'} == 0);
}
sub _NumSeq_X_neg_increasing_from_i {
my ($self) = @_;
### SquareSpiral _NumSeq_X_neg_increasing_from_i(): $self
# wider=0 from X=0
# wider=1 from X=-1
# wider=2 from X=-1
return int(($self->{'wider'}+1)/2);
}
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
sub _NumSeq_X_neg_min { # not the value at X=0,Y=0 if wider>0
my ($self) = @_;
return $self->n_start;
}
use constant _NumSeq_N_oeis_anum =>
{ 'wider=0,n_start=1' =>
{ X_axis => 'A054552', # spoke E, 4n^2 - 3n + 1
Y_neg => 'A033951', # spoke S, 4n^2 + 3n + 1
Diagonal_NW => 'A053755', # 4n^2 + 1
Diagonal_SE => 'A016754', # (2n+1)^2
# OEIS-Catalogue: A054552 planepath=SquareSpiral
# OEIS-Catalogue: A033951 planepath=SquareSpiral line_type=Y_neg
# OEIS-Catalogue: A053755 planepath=SquareSpiral line_type=Diagonal_NW
# OEIS-Catalogue: A016754 planepath=SquareSpiral line_type=Diagonal_SE
#
# OEIS-Other: A054552 planepath=GreekKeySpiral,turns=0
# OEIS-Other: A033951 planepath=GreekKeySpiral,turns=0 line_type=Y_neg
# OEIS-Other: A053755 planepath=GreekKeySpiral,turns=0 line_type=Diagonal_NW
# OEIS-Other: A016754 planepath=GreekKeySpiral,turns=0 line_type=Diagonal_SE
# Not quite, these have OFFSET=1 whereas path start X=0
# # Y_axis => 'A054556', # spoke N
# # X_neg => 'A054567', # spoke W
# # Diagonal => 'A054554', # spoke NE
# # Diagonal_SW => 'A054569', # spoke NE
# # # OEIS-Catalogue: A054556 planepath=SquareSpiral line_type=Y_axis
# # # OEIS-Catalogue: A054554 planepath=SquareSpiral line_type=Diagonal
},
'wider=0,n_start=0' =>
{ X_axis => 'A001107',
Y_axis => 'A033991',
Y_neg => 'A033954', # second 10-gonals
Diagonal => 'A002939',
Diagonal_NW => 'A016742', # 10-gonals average, 4*n^2
Diagonal_SW => 'A002943',
# OEIS-Other: A001107 planepath=SquareSpiral,n_start=0
# OEIS-Catalogue: A033991 planepath=SquareSpiral,n_start=0 line_type=Y_axis
# OEIS-Other: A033954 planepath=SquareSpiral,n_start=0 line_type=Y_neg
# OEIS-Catalogue: A002939 planepath=SquareSpiral,n_start=0 line_type=Diagonal
# OEIS-Other: A016742 planepath=SquareSpiral,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A002943 planepath=SquareSpiral,n_start=0 line_type=Diagonal_SW
},
'wider=1,n_start=1' =>
{ Diagonal_SW => 'A069894',
# OEIS-Catalogue: A069894 planepath=SquareSpiral,wider=1 line_type=Diagonal_SW
},
'wider=1,n_start=0' =>
{ Diagonal_SW => 'A016754', # odd squares
# OEIS-Other: A016754 planepath=SquareSpiral,wider=1,n_start=0 line_type=Diagonal_SW
},
};
}
{ package Math::PlanePath::GreekKeySpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
sub _NumSeq_X_neg_increasing {
my ($self) = @_;
return ($self->{'turns'} == 0); # when SquareSpiral style
}
*_NumSeq_Y_neg_increasing = \&_NumSeq_X_neg_increasing;
sub _NumSeq_Diagonal_increasing {
my ($self) = @_;
return ($self->{'turns'} <= 1);
}
sub _NumSeq_Diagonal_NW_increasing {
my ($self) = @_;
return ($self->{'turns'} == 0);
}
*_NumSeq_Diagonal_SW_increasing = \&_NumSeq_Diagonal_increasing;
sub _NumSeq_Diagonal_SE_increasing {
my ($self) = @_;
return ($self->{'turns'} <= 2);
}
use constant _NumSeq_N_oeis_anum =>
{ 'turns=0' =>
(Math::PlanePath::SquareSpiral
->_NumSeq_N_oeis_anum->{'wider=0,n_start=1'}
|| die "Oops, SquareSpiral NumSeq PlanePathN not found"),
};
}
{ package Math::PlanePath::PyramidSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A054552', # square spiral spoke E, 4n^2 - 3n + 1
Diagonal_SE => 'A033951', # square spiral spoke S, 4n^2 + 3n + 1
# OEIS-Other: A054552 planepath=PyramidSpiral
# OEIS-Other: A033951 planepath=PyramidSpiral line_type=Diagonal_SE
},
'n_start=0' =>
{ X_axis => 'A001107', # decagonal
Y_axis => 'A002939',
X_neg => 'A033991',
Y_neg => 'A002943',
Diagonal_SW => 'A007742',
Diagonal_SE => 'A033954', # decagonal second kind
# OEIS-Other: A001107 planepath=PyramidSpiral,n_start=0
# OEIS-Other: A002939 planepath=PyramidSpiral,n_start=0 line_type=Y_axis
# OEIS-Other: A033991 planepath=PyramidSpiral,n_start=0 line_type=X_neg
# OEIS-Other: A002943 planepath=PyramidSpiral,n_start=0 line_type=Y_neg
# OEIS-Other: A007742 planepath=PyramidSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Other: A033954 planepath=PyramidSpiral,n_start=0 line_type=Diagonal_SE
},
'n_start=2' =>
{ Diagonal_SE => 'A185669',
# OEIS-Catalogue: A185669 planepath=PyramidSpiral,n_start=2 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::TriangleSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A117625', # step by 2 each time
Y_neg => 'A006137', # step by 2 each time
Diagonal_SW => 'A064225',
Diagonal_SE => 'A081267',
# OEIS-Other: A117625 planepath=TriangleSpiral
# OEIS-Other: A064225 planepath=TriangleSpiral line_type=Diagonal_SW
# OEIS-Other: A081267 planepath=TriangleSpiral line_type=Diagonal_SE
# # Not quite, starts value=3 at n=0 which is path Y=1
# Diagonal => 'A064226', # and duplicate in A081269
},
'n_start=0' =>
{ Y_axis => 'A062741', # 3*pentagonal, Y even
Diagonal => 'A062708', # reading in direction 0,2,...
Diagonal_SW => 'A062725', # reading in direction 0,7,...
Diagonal_SE => 'A062728', # 11-gonal "second" per Math::NumSeq::Polygonal
# OEIS-Catalogue: A062741 planepath=TriangleSpiral,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A062708 planepath=TriangleSpiral,n_start=0 line_type=Diagonal
# OEIS-Catalogue: A062725 planepath=TriangleSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Other: A062728 planepath=TriangleSpiral,n_start=0 line_type=Diagonal_SE
# but spaced 2 apart ...
# X_axis => 'A051682', # 11-gonals per Math::NumSeq::Polygonal
# # OEIS-Other: A051682 planepath=TriangleSpiral,n_start=0 # X_axis
},
};
}
{ package Math::PlanePath::TriangleSpiralSkewed;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# ENHANCE-ME: All these variously rotated for skew=right,up,down
use constant _NumSeq_N_oeis_anum =>
{ 'skew=left,n_start=1' =>
{ X_axis => 'A117625',
X_neg => 'A006137',
Y_neg => 'A064225',
Diagonal => 'A081589',
Diagonal_SW => 'A038764',
Diagonal_SE => 'A081267',
# OEIS-Catalogue: A117625 planepath=TriangleSpiralSkewed
# OEIS-Catalogue: A006137 planepath=TriangleSpiralSkewed line_type=X_neg
# OEIS-Catalogue: A064225 planepath=TriangleSpiralSkewed line_type=Y_neg
# OEIS-Catalogue: A081589 planepath=TriangleSpiralSkewed line_type=Diagonal
# OEIS-Catalogue: A038764 planepath=TriangleSpiralSkewed line_type=Diagonal_SW
# OEIS-Catalogue: A081267 planepath=TriangleSpiralSkewed line_type=Diagonal_SE
# OEIS-Catalogue: A081274 planepath=TriangleSpiralSkewed line_type=Diagonal_SW
# # Not quite, starts OFFSET=0 value=3 but that is at path Y=1
# Y_axis => 'A064226', # and duplicate in A081269
},
'skew=left,n_start=0' =>
{ X_axis => 'A051682', # 11-gonals per Math::NumSeq::Polygonal
Y_axis => 'A062708', # reading in direction 0,2,...
Y_neg => 'A062725', # reading in direction 0,7,...
Diagonal_SE => 'A062728', # 11-gonal "second" per Math::NumSeq::Polygonal
Diagonal_SW => 'A081266',
# OEIS-Other: A051682 planepath=TriangleSpiralSkewed,n_start=0 # X_axis
# OEIS-Other: A062708 planepath=TriangleSpiralSkewed,n_start=0 line_type=Y_axis
# OEIS-Other: A062725 planepath=TriangleSpiralSkewed,n_start=0 line_type=Y_neg
# OEIS-Other: A062728 planepath=TriangleSpiralSkewed,n_start=0 line_type=Diagonal_SE
# OEIS-Catalogue: A081266 planepath=TriangleSpiralSkewed,n_start=0 line_type=Diagonal_SW
},
};
}
{ package Math::PlanePath::DiamondSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A130883', # 2*n^2-n+1
X_neg => 'A084849', # 2*n^2+n+1
Y_axis => 'A058331', # 2*n^2 + 1
Y_neg => 'A001844', # centred squares 2n(n+1)+1
# OEIS-Catalogue: A130883 planepath=DiamondSpiral
# OEIS-Other: A084849 planepath=DiamondSpiral line_type=X_neg
# OEIS-Catalogue: A058331 planepath=DiamondSpiral line_type=Y_axis
# OEIS-Other: A001844 planepath=DiamondSpiral line_type=Y_neg
},
'n_start=0' =>
{ X_axis => 'A000384', # 2*n^2-n, hexagonal numbers
X_neg => 'A014105', # 2*n^2+n, hexagonal numbers second kind
Y_axis => 'A001105', # 2*n^2
Y_neg => 'A046092', # 2n(n+1) = 4*triangular
# OEIS-Other: A000384 planepath=DiamondSpiral,n_start=0
# OEIS-Other: A014105 planepath=DiamondSpiral,n_start=0 line_type=X_neg
# OEIS-Other: A001105 planepath=DiamondSpiral,n_start=0 line_type=Y_axis
# OEIS-Other: A046092 planepath=DiamondSpiral,n_start=0 line_type=Y_neg
},
};
}
{ package Math::PlanePath::DiamondArms;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::AztecDiamondRings;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A001844', # centred squares 2n(n+1)+1
# OEIS-Other: A001844 planepath=AztecDiamondRings
# Not quite, A000384 has extra value=0
# Y_axis => 'A000384', # hexagonal numbers
},
'n_start=0' =>
{ X_axis => 'A046092', # 4*triangular
Diagonal => 'A139277',
# OEIS-Other: A046092 planepath=AztecDiamondRings,n_start=0
# OEIS-Other: A139277 planepath=AztecDiamondRings,n_start=0 line_type=Diagonal
},
};
}
{ package Math::PlanePath::PentSpiral;
use constant _NumSeq_X_axis_step => 2;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A192136', # (5*n^2-3*n+2)/2
X_neg => 'A116668', # (5n^2 + n + 2)/2
Diagonal_SE => 'A005891', # centred pentagonal (5n^2+5n+2)/2
# OEIS-Other: A192136 planepath=PentSpiral
# OEIS-Other: A116668 planepath=PentSpiral line_type=X_neg
# OEIS-Other: A005891 planepath=PentSpiral line_type=Diagonal_SE
# Not quite, A134238 OFFSET=1 vs start X=0 here
# Diagonal_SW => 'A134238',
},
'n_start=0' =>
{ X_axis => 'A000566', # heptagonals
Y_axis => 'A005476',
Diagonal_SE => 'A028895', # 5*triangular
# OEIS-Other: A000566 planepath=PentSpiral,n_start=0
# OEIS-Other: A005476 planepath=PentSpiral,n_start=0 line_type=Y_axis
# OEIS-Other: A028895 planepath=PentSpiral,n_start=0 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::PentSpiralSkewed;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_axis => 'A192136', # (5*n^2-3*n+2)/2
X_neg => 'A116668', # (5n^2 + n + 2)/2
Diagonal_NW => 'A158187', # 10*n^2 + 1
Diagonal_SE => 'A005891', # centred pentagonal (5n^2+5n+2)/2
# OEIS-Catalogue: A192136 planepath=PentSpiralSkewed
# OEIS-Catalogue: A116668 planepath=PentSpiralSkewed line_type=X_neg
# OEIS-Catalogue: A158187 planepath=PentSpiralSkewed line_type=Diagonal_NW
# OEIS-Catalogue: A005891 planepath=PentSpiralSkewed line_type=Diagonal_SE
# Not quite, A140066 OFFSET=1 but path start Y=0 here
# Y_axis => 'A140066', # (5n^2-11n+8)/2 but from Y=0 so using (n-1)
# Not quite, A134238 OFFSET=1 but path start Y=0 here
# Y_neg => 'A134238',
# # OEIS-Catalogue: A134238 planepath=PentSpiralSkewed line_type=Y_neg
},
'n_start=0' =>
{ X_axis => 'A000566', # heptagonals
Y_axis => 'A005476',
X_neg => 'A005475',
Diagonal_NW => 'A033583', # 10*n^2
Diagonal_SE => 'A028895', # 5*triangular
# OEIS-Other: A000566 planepath=PentSpiralSkewed,n_start=0
# OEIS-Catalogue: A005476 planepath=PentSpiralSkewed,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A005475 planepath=PentSpiralSkewed,n_start=0 line_type=X_neg
# OEIS-Other: A033583 planepath=PentSpiralSkewed,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A028895 planepath=PentSpiralSkewed,n_start=0 line_type=Diagonal_SE
# Not quite, A147875 OFFSET=1 vs start Y=0 here
# Y_neg => 'A147875', # second heptagonals
# # OEIS-Other: A147875 planepath=PentSpiralSkewed,n_start=0 line_type=Y_neg
},
};
}
{ package Math::PlanePath::HexSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
*_NumSeq_X_neg_increasing
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing;
*_NumSeq_X_neg_increasing_from_i
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i;
*_NumSeq_X_neg_min
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'wider=0,n_start=1' =>
{ X_axis => 'A056105', # first spoke 3n^2-2n+1
Diagonal => 'A056106', # second spoke 3n^2-n+1
Diagonal_NW => 'A056107', # third spoke 3n^2+1
X_neg => 'A056108', # fourth spoke 3n^2+n+1
Diagonal_SW => 'A056109', # fifth spoke 3n^2+2n+1
Diagonal_SE => 'A003215', # centred hexagonal numbers
# OEIS-Other: A056105 planepath=HexSpiral
# OEIS-Other: A056106 planepath=HexSpiral line_type=Diagonal
# OEIS-Other: A056107 planepath=HexSpiral line_type=Diagonal_NW
# OEIS-Other: A056108 planepath=HexSpiral line_type=X_neg
# OEIS-Other: A056109 planepath=HexSpiral line_type=Diagonal_SW
# OEIS-Other: A003215 planepath=HexSpiral line_type=Diagonal_SE
},
'wider=0,n_start=0' =>
{ X_axis => 'A000567', # octagonal numbers
X_neg => 'A049451',
Diagonal => 'A049450',
Diagonal_NW => 'A033428', # octagonal numbers first,second average
Diagonal_SW => 'A045944', # octagonal numbers second
Diagonal_SE => 'A028896',
# OEIS-Other: A000567 planepath=HexSpiral,n_start=0
# OEIS-Other: A049451 planepath=HexSpiral,n_start=0 line_type=X_neg
# OEIS-Catalogue: A049450 planepath=HexSpiral,n_start=0 line_type=Diagonal
# OEIS-Other: A033428 planepath=HexSpiral,n_start=0 line_type=Diagonal_NW
# OEIS-Other: A045944 planepath=HexSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Catalogue: A028896 planepath=HexSpiral,n_start=0 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::HexSpiralSkewed;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
*_NumSeq_X_neg_increasing
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing;
*_NumSeq_X_neg_increasing_from_i
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i;
*_NumSeq_X_neg_min
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'wider=0,n_start=1' =>
{ X_axis => 'A056105', # first spoke 3n^2-2n+1
Y_axis => 'A056106', # second spoke 3n^2-n+1
Diagonal_NW => 'A056107', # third spoke 3n^2+1
X_neg => 'A056108', # fourth spoke 3n^2+n+1
Y_neg => 'A056109', # fifth spoke 3n^2+2n+1
Diagonal_SE => 'A003215', # centred hexagonal numbers
# OEIS-Catalogue: A056105 planepath=HexSpiralSkewed
# OEIS-Catalogue: A056106 planepath=HexSpiralSkewed line_type=Y_axis
# OEIS-Catalogue: A056108 planepath=HexSpiralSkewed line_type=X_neg
# OEIS-Catalogue: A056109 planepath=HexSpiralSkewed line_type=Y_neg
# OEIS-Catalogue: A056107 planepath=HexSpiralSkewed line_type=Diagonal_NW
# OEIS-Other: A003215 planepath=HexSpiralSkewed line_type=Diagonal_SE
},
'wider=0,n_start=0' =>
{ X_axis => 'A000567', # octagonal numbers
Y_axis => 'A049450',
X_neg => 'A049451',
Y_neg => 'A045944', # octagonal numbers second
Diagonal => 'A062783',
Diagonal_NW => 'A033428', # octagonal numbers first,second average
Diagonal_SW => 'A063436',
Diagonal_SE => 'A028896',
# OEIS-Other: A000567 planepath=HexSpiralSkewed,n_start=0
# OEIS-Other: A049450 planepath=HexSpiralSkewed,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A049451 planepath=HexSpiralSkewed,n_start=0 line_type=X_neg
# OEIS-Other: A045944 planepath=HexSpiralSkewed,n_start=0 line_type=Y_neg
# OEIS-Catalogue: A062783 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal
# OEIS-Other: A033428 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A063436 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_SW
# OEIS-Other: A028896 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_SE
},
# wider=1 X_axis almost 3*n^2 but not initial X=0 value
# wider=1 Y_axis almost A049451 twice pentagonal but not initial X=0
# wider=2 Y_axis almost A028896 6*triangular but not initial Y=0
};
}
{ package Math::PlanePath::HexArms;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::HeptSpiralSkewed;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
# 'n_start=1' =>
# {
# # Not quite, OFFSET=1 vs path start X=Y=0
# # Y_axis => 'A140065', # (7n^2 - 17n + 12)/2 but starting Y=0 not n=1
# # Diagonal_NW => 'A140063',
# # Diagonal_SE => 'A069099',
# },
'n_start=0' =>
{ X_axis => 'A001106', # 9-gonals
Y_axis => 'A218471',
X_neg => 'A022265',
Y_neg => 'A179986', # second 9-gonals
Diagonal => 'A195023',
Diagonal_NW => 'A022264',
Diagonal_SW => 'A186029',
Diagonal_SE => 'A024966',
# OEIS-Other: A001106 planepath=HeptSpiralSkewed,n_start=0
# OEIS-Catalogue: A022265 planepath=HeptSpiralSkewed,n_start=0 line_type=X_neg
# OEIS-Catalogue: A218471 planepath=HeptSpiralSkewed,n_start=0 line_type=Y_axis
# OEIS-Other: A179986 planepath=HeptSpiralSkewed,n_start=0 line_type=Y_neg
# OEIS-Catalogue: A195023 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal
# OEIS-Catalogue: A022264 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A186029 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_SW
# OEIS-Catalogue: A024966 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::OctagramSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
'n_start=1' =>
{ Diagonal_SE => 'A194268',
# OEIS-Other: A194268 planepath=OctagramSpiral line_type=Diagonal_SE
# Not quite, but A125201 doesn't have initial N=1 for path origin
# X_axis => 'A125201'
},
'n_start=0' =>
{ X_axis => 'A051870', # 18-gonals
Y_axis => 'A139273',
X_neg => 'A139275',
Y_neg => 'A139277',
Diagonal => 'A139272',
Diagonal_NW => 'A139274',
Diagonal_SW => 'A139276',
Diagonal_SE => 'A139278', # second 18-gonals
# OEIS-Other: A051870 planepath=OctagramSpiral,n_start=0
# OEIS-Catalogue: A139273 planepath=OctagramSpiral,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A139275 planepath=OctagramSpiral,n_start=0 line_type=X_neg
# OEIS-Catalogue: A139277 planepath=OctagramSpiral,n_start=0 line_type=Y_neg
# OEIS-Catalogue: A139272 planepath=OctagramSpiral,n_start=0 line_type=Diagonal
# OEIS-Catalogue: A139274 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A139276 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Other: A139278 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::AnvilSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
*_NumSeq_X_neg_increasing
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing;
*_NumSeq_X_neg_increasing_from_i
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i;
*_NumSeq_X_neg_min
= \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'wider=0,n_start=1' =>
{ X_axis => 'A033570', # odd pentagonals (2n+1)*(3n+1)
Diagonal => 'A033568', # odd second pentagonals
Diagonal_SE => 'A085473',
# OEIS-Catalogue: A033570 planepath=AnvilSpiral
# OEIS-Catalogue: A033568 planepath=AnvilSpiral line_type=Diagonal
# OEIS-Catalogue: A085473 planepath=AnvilSpiral line_type=Diagonal_SE
# Not quite, A136392 OFFSET=1 value=1 whereas path start Y=0 value=1
# Y_neg => 'A136392', # 1,9,29,61, 6n^2-10n+5
},
'wider=0,n_start=1,i_start=1' =>
{ Y_axis => 'A126587', # points within 3,4,5 triangle, starting value=3
# OEIS-Catalogue: A126587 planepath=AnvilSpiral line_type=Y_axis i_start=1
},
'wider=0,n_start=0' =>
{ X_axis => 'A211014', # 14-gonal second
Y_axis => 'A139267', # 2*octagonal
X_neg => 'A049452', # alternate pentagonals
Y_neg => 'A033580', # 4*second pentagonals
Diagonal => 'A051866', # 14-gonals
Diagonal_NW => 'A094159', # 3*hexagonal
Diagonal_SW => 'A049453',
Diagonal_SE => 'A195319', # 3*second hexagonal
# OEIS-Other: A211014 planepath=AnvilSpiral,n_start=0
# OEIS-Other: A051866 planepath=AnvilSpiral,n_start=0 line_type=Diagonal
# OEIS-Catalogue: A139267 planepath=AnvilSpiral,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A049452 planepath=AnvilSpiral,n_start=0 line_type=X_neg
# OEIS-Catalogue: A033580 planepath=AnvilSpiral,n_start=0 line_type=Y_neg
# OEIS-Catalogue: A094159 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_NW
# OEIS-Catalogue: A049453 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Catalogue: A195319 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_SE
},
# Not quite, A051866 starts value=0 which is at X=-1
# 'wider=1,n_start=0' =>
# { X_axis => 'A051866',
# # OEIS-Other: A051866 planepath=AnvilSpiral,wider=1,n_start=0
# },
# 'wider=1,n_start=0' =>
# { Diagonal_NW_minus_1 => 'A033569', # (2*n-1)*(3*n+1) starting -1
# # XX-Other: A033569 planepath=AnvilSpiral,wider=1,n_start=-1 line_type=Diagonal_NW
# },
# 'wider=2,n_start=1' =>
# {
# Not quite, A033581 initial value=2 whereas path start N=0
# # Y_axis => 'A033581', # 6*n^2 is 14-gonals pairs average in Math::NumSeq::Polygonal
# # # OEIS-Other: A033581 planepath=AnvilSpiral,wider=2 line_type=Y_axis
# },
};
}
{ package Math::PlanePath::KnightSpiral;
use constant _NumSeq_Diagonal_increasing => 1; # low then high
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::CretanLabyrinth;
use constant _NumSeq_X_axis_increasing => 1;
}
{ package Math::PlanePath::SquareArms;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::SacksSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# SacksSpiral X_axis -- squares (i-1)^2, starting from i=1 value=0
}
{ package Math::PlanePath::VogelFloret;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::TheodorusSpiral;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::ArchimedeanChords;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::MultipleRings;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::PixelRings;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # where covered
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::FilledRings;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# use constant _NumSeq_N_oeis_anum =>
# {
# # Not quite, A036704 OFFSET=0 value=1,9,21 vs X=0 value=0,1,9,21
# 'n_start=0' =>
# { X_axis => 'A036704', # count points norm <= n+1/2
# },
# };
}
{ package Math::PlanePath::Hypot;
sub _NumSeq_X_axis_i_start {
my ($self) = @_;
### _NumSeq_X_axis_i_start() ...
return ($self->{'points'} eq 'odd'
? 1 # X=0,Y=0 not visited
: 0);
}
sub _NumSeq_X_neg_i_start {
my ($self) = @_;
### _NumSeq_X_axis_i_start() ...
return ($self->{'points'} eq 'odd'
? -1 # X=0,Y=0 not visited
: 0);
}
*_NumSeq_Y_axis_i_start = \&_NumSeq_X_axis_i_start;
*_NumSeq_Y_neg_i_start = \&_NumSeq_X_neg_i_start;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'points=all,n_start=0' =>
{ X_axis => 'A051132', # count points < n^2
# OEIS-Catalogue: A051132 planepath=Hypot,n_start=0
},
};
}
{ package Math::PlanePath::HypotOctant;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
*_NumSeq_X_axis_i_start = \&Math::PlanePath::Hypot::_NumSeq_X_axis_i_start;
use constant _NumSeq_N_oeis_anum =>
{ 'points=even' =>
{ Diagonal => 'A036702', # count points |z|<=n for 0<=b<=a
# OEIS-Catalogue: A036702 planepath=HypotOctant,points=even line_type=Diagonal
},
};
}
{ package Math::PlanePath::TriangularHypot;
sub _NumSeq_X_axis_i_start {
my ($self) = @_;
return ($self->{'points'} eq 'odd' || $self->{'points'} eq 'hex_centred'
? 1 # X=0,Y=0 not visited
: 0);
}
*_NumSeq_Diagonal_i_start = \&_NumSeq_X_axis_i_start;
*_NumSeq_Diagonal_SE_i_start = \&_NumSeq_X_axis_i_start;
sub _NumSeq_X_neg_i_start {
my ($self) = @_;
return - $self->_NumSeq_X_axis_i_start;
}
*_NumSeq_Diagonal_NW_i_start = \&_NumSeq_X_axis_i_start;
*_NumSeq_Diagonal_SW_i_start = \&_NumSeq_X_axis_i_start;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::PythagoreanTree;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_N_oeis_all_anum =>
{ Depth_start => 'A007051', # (3^n+1)/2
# OEIS-Catalogue: A007051 planepath=PythagoreanTree line_type=Depth_start
# Not quite, Depth_end=(3^(n+1)-1)/2, so is n+1
# Depth_end => 'A003462', # (3^n-1)/2
};
}
{ package Math::PlanePath::RationalsTree;
use constant _NumSeq_X_axis_increasing => 1;
sub _NumSeq_Y_axis_increasing {
my ($self) = @_;
return ($self->{'tree_type'} eq 'L' ? 0 : 1);
}
sub _NumSeq_Y_axis_increasing_from_i {
my ($self) = @_;
return ($self->{'tree_type'} eq 'L' ? 2 : 1);
}
use constant _NumSeq_Y_axis_min => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'tree_type=SB' =>
{ Depth_start => 'A000079', # powers-of-2
# RationalsTree SB -- X_axis 2^n-1 but starting X=1
# RationalsTree SB,CW -- Y_axis A000079 2^n but starting Y=1
},
'tree_type=CW' =>
{ Depth_start => 'A000079', # powers-of-2
},
'tree_type=Bird' =>
{ X_axis => 'A081254', # local max sumdisttopow2(m)/m^2
Depth_start => 'A000079', # powers-of-2
# OEIS-Catalogue: A081254 planepath=RationalsTree,tree_type=Bird
# OEIS-Other: A000079 planepath=RationalsTree,tree_type=Bird line_type=Depth_start
# RationalsTree Bird -- Y_axis almost A000975 10101 101010 no
# consecutive equal bits, but start=1
},
'tree_type=Drib' =>
{ X_axis => 'A086893', # pos of fibonacci F(n+1)/F(n) in Stern diatomic
Depth_start => 'A000079', # powers-of-2
# OEIS-Catalogue: A086893 planepath=RationalsTree,tree_type=Drib
# Drib Y_axis
# Not quite, A061547 OFFSET=1 value=0 cf path Y=1 value N=1
# Y_axis => 'A061547'# derangements or alternating bits plus pow4
},
'tree_type=AYT' =>
{ Depth_start => 'A000079', # powers-of-2
# RationalsTree AYT -- Y_axis A083318 2^n+1 but starting Y=1
},
'tree_type=HCS' =>
{ Depth_start => 'A000079', # powers-of-2
# RationalsTree HCS
# Not quite, A000079 OFFSET=0 value=1 cf here X=1 N=1
# X_axis => 'A000079', # powers 2^X
# Not quite, A007283 OFFSET=0 and doesn't have extra N=1 at Y=1
# Y_axis => 'A007283', # 3*2^n starting OFFSET=0 value=3
},
};
}
{ package Math::PlanePath::FractionsTree;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_X_offset => -1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Y_axis_i_start => 2;
}
{ package Math::PlanePath::ChanTree;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_X_axis_i_start => 1;
use constant _NumSeq_Y_axis_i_start => 1; # start at Y=1
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'k=2,n_start=1' =>
{ Depth_start => 'A000079', # powers-of-2
# OEIS-Other: A000079 planepath=ChanTree,n_start=1,k=2 line_type=Depth_start
# Depth_end is 2^k-1 A000225, or 2^k-2 A000918, but without initial
# 0 or -1.
},
'k=2,n_start=0' =>
{ Depth_start => 'A000225', # 2^k-1
# OEIS-Other: A000225 planepath=ChanTree,k=2 line_type=Depth_start
},
'k=3,n_start=1' =>
{ Depth_start => 'A000244', # powers-of-3
# OEIS-Other: A000244 planepath=ChanTree,n_start=1 line_type=Depth_start
},
'k=4,n_start=1' =>
{ Depth_start => 'A000302', # powers-of-4
# OEIS-Other: A000302 planepath=ChanTree,n_start=1,k=4 line_type=Depth_start
},
'k=5,n_start=1' =>
{ Depth_start => 'A000351', # powers-of-5
# OEIS-Other: A000351 planepath=ChanTree,n_start=1,k=5 line_type=Depth_start
},
'k=10,n_start=1' =>
{ Depth_start => 'A011557', # powers-of-10
# OEIS-Other: A011557 planepath=ChanTree,n_start=1,k=10 line_type=Depth_start
},
};
}
{ package Math::PlanePath::DiagonalRationals;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_axis_i_start => 1;
use constant _NumSeq_Y_axis_i_start => 1;
# Diagonal => 'A002088', # cumulative totient
# Not quite, start X=1 value=1 cf seq OFFSET=0 value=0
}
{ package Math::PlanePath::FactorRationals;
use constant _NumSeq_X_axis_i_start => 1;
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
# perfect squares along X axis of even/odd
return $self->{'factor_coding'} eq 'even/odd';
}
use constant _NumSeq_Y_axis_i_start => 1;
sub _NumSeq_Y_axis_increasing {
my ($self) = @_;
# perfect squares along Y axis of odd/even
return $self->{'factor_coding'} eq 'odd/even';
}
use constant _NumSeq_N_oeis_anum =>
{ 'factor_coding=even/odd' =>
{ Y_axis => 'A102631', # n^2/(squarefree kernel)
# OEIS-Catalogue: A102631 planepath=FactorRationals line_type=Y_axis
# # Not quite, OFFSET=0 value 0 whereas start X=Y=1 value 1 here
# X_axis => 'A000290', # squares 0,1,4,9
# # OEIS-Other: A000290 planepath=FactorRationals
},
'factor_coding=odd/even' =>
{ X_axis => 'A102631', # n^2/(squarefree kernel)
# OEIS-Other: A102631 planepath=FactorRationals,factor_coding=odd/even
},
};
}
{ package Math::PlanePath::GcdRationals;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_axis_i_start => 1;
use constant _NumSeq_Y_axis_i_start => 1;
# GcdRationals
# Not quite, starts X=1
# X_axis => triangular row
# Not quite, starts X=1 here cf OFFSET=0 in A000124
# Y_axis => 'A000124', # triangular+1
#
# GcdRationals,pairs_order=diagonals_down
# Not quite, start X=1 here cf A000290 starts OFFSET=1
# X_axis => 'A000290', # Y=1 row, perfect squares
# Not quite, A033638 starts two ones 1,1,...
# Y_axis => 'A033638', # quarter-squares + 1
#
# GcdRationals,pairs_order=diagonals_up
# Not quite, A002061 starts two ones 1,1,
# X_axis => 'A002061',
# Not quite, X=1 column squares+pronic, but no initial 0,0 of A002620
# Y_axis => 'A002620', # X=1 column
# Not quite, starting value=2 here
# Diagonal_above => 'A002522', # Y=X+1 diagonal, squares+1
}
{ package Math::PlanePath::CfracDigits;
# 1
# diagonal Y/(Y+1) = 0 + -----
# 1 + 1/Y
# q0=1 q1=Y
# N = 3,Y-1 in 1,2,3
# = 1,0,Y-1 in 0,1,2
#
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_X_offset => -1;
use constant _NumSeq_Y_axis_increasing => 1; # radix without digit 0
}
{ package Math::PlanePath::PeanoCurve;
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
return ($self->{'radix'} % 2);
}
*_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing;
use constant _NumSeq_N_oeis_anum =>
{ 'radix=3' =>
{ X_axis => 'A163480', # axis same as initial direction
Y_axis => 'A163481', # axis opp to initial direction
Diagonal => 'A163343',
},
# OEIS-Catalogue: A163480 planepath=PeanoCurve
# OEIS-Catalogue: A163481 planepath=PeanoCurve line_type=Y_axis
# OEIS-Catalogue: A163343 planepath=PeanoCurve line_type=Diagonal
# OEIS-Other: A163480 planepath=GrayCode,apply_type=TsF,radix=3
# OEIS-Other: A163481 planepath=GrayCode,apply_type=TsF,radix=3 line_type=Y_axis
# OEIS-Other: A163343 planepath=GrayCode,apply_type=TsF,radix=3 line_type=Diagonal
# OEIS-Other: A163480 planepath=GrayCode,apply_type=FsT,radix=3
# OEIS-Other: A163481 planepath=GrayCode,apply_type=FsT,radix=3 line_type=Y_axis
# OEIS-Other: A163343 planepath=GrayCode,apply_type=FsT,radix=3 line_type=Diagonal
};
}
{ package Math::PlanePath::WunderlichSerpentine;
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
if ($self->{'radix'} % 2) {
return 1; # odd radix always increasing
}
# FIXME: depends on the serpentine_type bits
return 0;
}
sub _NumSeq_Y_axis_increasing {
my ($self) = @_;
if ($self->{'radix'} % 2) {
return 1; # odd radix always increasing
}
# FIXME: depends on the serpentine_type bits
return 0;
}
}
{ package Math::PlanePath::HilbertCurve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ '' =>
{ X_axis => 'A163482',
Y_axis => 'A163483',
Diagonal => 'A062880', # base 4 digits 0,2 only
# OEIS-Catalogue: A163482 planepath=HilbertCurve
# OEIS-Catalogue: A163483 planepath=HilbertCurve line_type=Y_axis
# OEIS-Other: A062880 planepath=HilbertCurve line_type=Diagonal
},
};
}
{ package Math::PlanePath::HilbertSpiral;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
}
{ package Math::PlanePath::ZOrderCurve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'radix=2' =>
{ X_axis => 'A000695', # base 4 digits 0,1 only (RadixConversion)
Y_axis => 'A062880', # base 4 digits 0,2 only
Diagonal => 'A001196', # base 4 digits 0,3 only
# OEIS-Other: A000695 planepath=ZOrderCurve
# OEIS-Catalogue: A062880 planepath=ZOrderCurve line_type=Y_axis
# OEIS-Catalogue: A001196 planepath=ZOrderCurve line_type=Diagonal
},
'radix=3,i_start=1' =>
{ X_axis => 'A037314', # base 9 digits 0,1,2 only, starting OFFSET=1 value=1
Y_axis => 'A208665', # base 9 digits 0,3,6 only, starting OFFSET=1 value=3
# OEIS-Catalogue: A037314 planepath=ZOrderCurve,radix=3 i_start=1
# OEIS-Catalogue: A208665 planepath=ZOrderCurve,radix=3 i_start=1 line_type=Y_axis
# ZOrderCurve dir=2 radix,3: match 6,27,30,33,54,57,60,243,246,249,270,273,276,297,300
# A208665 Numbers that match odd ternary polynomials; see Comments.
# A208665 ,3,6,27,30,33,54,57,60,243,246,249,270,273,276,297,300,303,486,489,492,513,516,519,540,543,546,2187,2190,2193,2214,2217,2220,2241,2244,2247,2430,2433,2436,2457,2460,2463,2484,2487,2490,2673,2676,2679,
# base 9 digits 0,3,6 only
},
'radix=10' =>
{ X_axis => 'A051022', # base 10 insert 0s, for digits 0 to 9 base 100
# OEIS-Catalogue: A051022 planepath=ZOrderCurve,radix=10
},
};
}
{ package Math::PlanePath::GrayCode;
# X axis increasing for:
# radix=2 TsF,Fs
# radix=3 reflected TsF,FsT
# radix=3 modular TsF,Fs
# radix=4 reflected TsF,Fs
# radix=4 modular TsF,Fs
# radix=5 reflected TsF,FsT
# radix=5 modular TsF,Fs
#
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
if ($self->{'gray_type'} eq 'modular' || $self->{'radix'} == 2) {
return ($self->{'apply_type'} eq 'TsF'
|| $self->{'apply_type'} eq 'Fs');
}
if ($self->{'radix'} & 1) {
return ($self->{'apply_type'} eq 'TsF'
|| $self->{'apply_type'} eq 'FsT');
} else {
return ($self->{'apply_type'} eq 'TsF'
|| $self->{'apply_type'} eq 'Fs');
}
}
*_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing;
# Diagonal increasing for:
# radix=2 FsT,Ts
# radix=3 reflected Ts,Fs
# radix=3 modular FsT
# radix=4 reflected FsT,Ts
# radix=4 modular FsT
# radix=5 reflected Ts,Fs
# radix=5 modular FsT
sub _NumSeq_Diagonal_increasing {
my ($self) = @_;
if ($self->{'radix'} & 1) {
if ($self->{'gray_type'} eq 'modular') {
return ($self->{'apply_type'} eq 'FsT'); # odd modular
} else {
return ($self->{'apply_type'} eq 'Ts'
|| $self->{'apply_type'} eq 'Fs'); # odd reflected
}
}
if ($self->{'gray_type'} eq 'reflected' || $self->{'radix'} == 2) {
return ($self->{'apply_type'} eq 'FsT'
|| $self->{'apply_type'} eq 'Ts'); # even reflected
} else {
return ($self->{'apply_type'} eq 'FsT'); # even modular
}
}
use constant _NumSeq_N_oeis_anum =>
{
'apply_type=TsF,gray_type=reflected,radix=3' =>
(Math::PlanePath::PeanoCurve->_NumSeq_N_oeis_anum->{'radix=3'}
|| die "Oops, SquareSpiral NumSeq PlanePathN not found"),
'apply_type=FsT,gray_type=reflected,radix=3' =>
(Math::PlanePath::PeanoCurve->_NumSeq_N_oeis_anum->{'radix=3'}
|| die "Oops, SquareSpiral NumSeq PlanePathN not found"),
# GrayCode radix=2 TsF==Fs reflected==modular
do {
my $href =
{ Y_axis => 'A001196', # base 4 digits 0,3 only
};
('apply_type=TsF,gray_type=reflected,radix=2' => $href,
'apply_type=Fs,gray_type=reflected,radix=2' => $href,
'apply_type=TsF,gray_type=modular,radix=2' => $href,
'apply_type=Fs,gray_type=modular,radix=2' => $href,
);
# OEIS-Other: A001196 planepath=GrayCode,apply_type=TsF line_type=Y_axis
# OEIS-Other: A001196 planepath=GrayCode,apply_type=Fs line_type=Y_axis
# OEIS-Other: A001196 planepath=GrayCode,apply_type=TsF,gray_type=modular line_type=Y_axis
# OEIS-Other: A001196 planepath=GrayCode,apply_type=Fs,gray_type=modular line_type=Y_axis
},
# GrayCode radix=2 Ts==FsT reflected==modular
do {
my $href =
{ Diagonal => 'A062880', # base 4 digits 0,2 only
};
('apply_type=Ts,gray_type=reflected,radix=2' => $href,
'apply_type=Ts,gray_type=modular,radix=2' => $href,
'apply_type=FsT,gray_type=reflected,radix=2' => $href,
'apply_type=FsT,gray_type=modular,radix=2' => $href,
);
# OEIS-Other: A062880 planepath=GrayCode,apply_type=Ts line_type=Diagonal
# OEIS-Other: A062880 planepath=GrayCode,apply_type=Ts,gray_type=modular line_type=Diagonal
# OEIS-Other: A062880 planepath=GrayCode,apply_type=FsT line_type=Diagonal
# OEIS-Other: A062880 planepath=GrayCode,apply_type=FsT,gray_type=modular line_type=Diagonal
},
# GrayCode radix=3 sT==sF reflected
# N split then toGray giving Y=0 means N ternary 010202 etc
# N split then toGray giving X=Y means N ternary pairs 112200
do {
my $href =
{ X_axis => 'A163344', # central Peano/4, base9 digits 0,1,2 only
Diagonal => 'A163343', # central diagonal of Peano, base9 0,4,8
};
('apply_type=sT,gray_type=reflected,radix=3' => $href,
'apply_type=sF,gray_type=reflected,radix=3' => $href,
);
# OEIS-Catalogue: A163344 planepath=GrayCode,apply_type=sT,radix=3
# OEIS-Other: A163344 planepath=GrayCode,apply_type=sF,radix=3
# OEIS-Other: A163343 planepath=GrayCode,apply_type=sT,radix=3 line_type=Diagonal
# OEIS-Other: A163343 planepath=GrayCode,apply_type=sF,radix=3 line_type=Diagonal
},
'apply_type=FsT,gray_type=modular,radix=3,i_start=1' =>
{ Diagonal => 'A208665', # base 9 digits 0,3,6
# OEIS-Other: A208665 planepath=GrayCode,apply_type=FsT,gray_type=modular,radix=3 line_type=Diagonal i_start=1
},
};
}
# { package Math::PlanePath::ImaginaryBase;
# }
{ package Math::PlanePath::ImaginaryHalf;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'radix=2,digit_order=YXX' =>
{ Y_axis => 'A033045', # base 8 digits 0,1 only
# OEIS-Other: A033045 planepath=ImaginaryHalf,digit_order=YXX line_type=Y_axis
},
'radix=2,digit_order=YXnX' =>
{ Y_axis => 'A033045', # base 8 digits 0,1 only
# OEIS-Other: A033045 planepath=ImaginaryHalf,digit_order=YXnX line_type=Y_axis
}
};
}
# { package Math::PlanePath::CubicBase;
# }
{ package Math::PlanePath::DekkingCurve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
}
{ package Math::PlanePath::DekkingCentres;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
}
{ package Math::PlanePath::CincoCurve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
}
{ package Math::PlanePath::BetaOmega;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
}
{ package Math::PlanePath::KochelCurve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
}
{ package Math::PlanePath::AR2W2Curve;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
}
{ package Math::PlanePath::WunderlichMeander;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
}
# { package Math::PlanePath::Flowsnake;
# }
# { package Math::PlanePath::FlowsnakeCentres;
# # inherit from Flowsnake
# }
# { package Math::PlanePath::GosperIslands;
# }
# { package Math::PlanePath::GosperSide;
# }
{ package Math::PlanePath::KochCurve;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_Diagonal_increasing => 1; # when touched
}
{ package Math::PlanePath::KochPeaks;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1; # when touched
# Diagonal never touched
}
{ package Math::PlanePath::KochSnowflakes;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::KochSquareflakes;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::QuadricCurve;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Diagonal_increasing => 1; # two values only
}
{ package Math::PlanePath::QuadricIslands;
# FIXME: pred() on Diagonal_SW doesn't notice 0.5 square
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 0;
use constant _NumSeq_Y_neg_increasing_from_i => 1; # after 3,2,8
use constant _NumSeq_Y_neg_min => 2; # at X=-1,Y=0 rather than X=0,Y=0
use constant _NumSeq_Diagonal_SW_increasing => 0;
use constant _NumSeq_Diagonal_SW_min => 1;
}
{ package Math::PlanePath::SierpinskiTriangle;
use constant _NumSeq_X_axis_increasing => 1; # for "diagonal" style
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
# low 10111=23 increment to 11000=24
# 10111 ones=4 width=2^4
# cf A160722 is 3*A006046-2*n, drawing three Sierpinski triangles
# http://www.polprimos.com/imagenespub/polca722.jpg
#
use constant _NumSeq_N_oeis_anum =>
{
#---------
# i_start=0, n_start=0
'align=triangular,n_start=0' =>
{ Diagonal_NW => 'A006046',
Depth_start => 'A006046',
# OEIS-Other: A006046 planepath=SierpinskiTriangle line_type=Diagonal_NW
# OEIS-Other: A006046 planepath=SierpinskiTriangle line_type=Depth_start
},
'align=right,n_start=0' =>
{ Y_axis => 'A006046',
Depth_start => 'A006046',
# OEIS-Catalogue: A006046 planepath=SierpinskiTriangle,align=diagonal,n_start=0 line_type=Y_axis
},
'align=left,n_start=0' =>
{ Diagonal_NW => 'A006046',
Depth_start => 'A006046',
# OEIS-Other: A006046 planepath=SierpinskiTriangle,align=left,n_start=0 line_type=Diagonal_NW
},
'align=diagonal,n_start=0' =>
{ Y_axis => 'A006046',
Depth_start => 'A006046',
# OEIS-Other: A006046 planepath=SierpinskiTriangle,align=diagonal,n_start=0 line_type=Y_axis
},
#---------
# i_start=1, n_start=0
# starting OFFSET=1 value=2,4,8,10 so missing N=0 at Y=0, hence i_start=1
'align=triangular,n_start=0,i_start=1' =>
{ Diagonal => 'A074330',
Depth_end => 'A074330',
# OEIS-Catalogue: A074330 planepath=SierpinskiTriangle line_type=Diagonal i_start=1
# OEIS-Other: A074330 planepath=SierpinskiTriangle line_type=Depth_end i_start=1
},
'align=right,n_start=0,i_start=1' =>
{ Diagonal => 'A074330',
Depth_end => 'A074330',
# OEIS-Other: A074330 planepath=SierpinskiTriangle,align=right line_type=Diagonal i_start=1
},
'align=left,n_start=0,i_start=1' =>
{ Y_axis => 'A074330',
Depth_end => 'A074330',
# OEIS-Other: A074330 planepath=SierpinskiTriangle,align=left line_type=Y_axis i_start=1
},
'align=diagonal,n_start=0,i_start=1' =>
{ X_axis => 'A074330',
Depth_end => 'A074330',
# OEIS-Other: A074330 planepath=SierpinskiTriangle,align=diagonal i_start=1
},
};
}
{ package Math::PlanePath::SierpinskiArrowhead;
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
# align="diagonal" is X increasing, other align is single origin point only
use constant _NumSeq_X_axis_increasing => 1;
}
{ package Math::PlanePath::SierpinskiArrowheadCentres;
use constant _NumSeq_Y_axis_increasing => 1; # never touched ?
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
# align="diagonal" is X increasing, other align is single origin point only
use constant _NumSeq_X_axis_increasing => 1;
}
{ package Math::PlanePath::SierpinskiCurve;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_X_axis_i_start => 1; # but not all cells visited
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_i_start => 1; # but not all cells visited
use constant _NumSeq_X_neg_increasing => 1; # arms
use constant _NumSeq_Y_neg_increasing => 1; # arms
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::SierpinskiCurveStair;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_X_neg_increasing => 1; # arms
use constant _NumSeq_Y_neg_increasing => 1; # arms
use constant _NumSeq_Diagonal_increasing => 1; # when touched
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
}
{ package Math::PlanePath::HIndexing;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when touched
}
# { package Math::PlanePath::DragonCurve;
# }
# { package Math::PlanePath::DragonRounded;
# }
# { package Math::PlanePath::DragonMidpoint;
# }
{ package Math::PlanePath::AlternatePaper;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1; # arms
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# selecting the smaller N on the negative axes gives increasing, maybe
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'arms=1' =>
{ X_axis => 'A000695', # base 4 digits 0,1 only
Diagonal => 'A062880', # base 4 digits 0,2 only
# OEIS-Other: A000695 planepath=AlternatePaper
# OEIS-Other: A062880 planepath=AlternatePaper line_type=Diagonal
},
# FIXME: not sure what to do when multiple-visited points on axes
# 'arms=2' =>
# { X_axis => 'A062880', # base 4 digits 0,2 only
# Y_axis => 'A145812', # base 4 digits 0,2 only except low digit 1,3 only
# # OEIS-Other: A062880 planepath=AlternatePaper,arms=2
# # OEIS-Catalogue: A145812 planepath=AlternatePaper,arms=2 line_type=Y_axis
# },
# 'arms=3' =>
# { X_axis => 'A001196', # base 4 digits 0,3 only
# # OEIS-Other: A001196 planepath=AlternatePaper,arms=3
# },
#
# alt paper arms=4 diagonal, arms=8 x axis
# A127988 # base 8 digits 0,4 only
};
}
{ package Math::PlanePath::AlternatePaperMidpoint;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1; # arms
use constant _NumSeq_Diagonal_SE_increasing => 1; # arms
}
# { package Math::PlanePath::TerdragonCurve;
# }
# { package Math::PlanePath::TerdragonRounded;
# }
# { package Math::PlanePath::TerdragonMidpoint;
# }
# { package Math::PlanePath::R5DragonCurve;
# }
# { package Math::PlanePath::R5DragonMidpoint;
# }
# { package Math::PlanePath::CCurve;
# }
# { package Math::PlanePath::ComplexPlus;
# }
{ package Math::PlanePath::ComplexMinus;
use constant _NumSeq_N_oeis_anum =>
{ 'realpart=1' =>
{ X_axis => 'A066321', # binary base i-1
# OEIS-Catalogue: A066321 planepath=ComplexMinus
# cf A066323 count of 1-bits in N on X axis
},
};
}
# { package Math::PlanePath::ComplexRevolving;
# }
{ package Math::PlanePath::Rows;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Y_neg_min => undef; # negatives
use constant _NumSeq_Y_neg_max => 1; # negatives
# secret negatives
# (w-1)*(w-1)-1
# = w^2-2w+1-1
# = w(w-2)
sub _NumSeq_Diagonal_SE_min {
my ($self) = @_;
return ($self->{'width'}-2)*$self->{'width'};
}
use constant _NumSeq_N_oeis_anum =>
{
'n_start=0,width=1' =>
{ X_axis => 'A001477', # integers 0,1,2,3,etc
# OEIS-Other: A001477 planepath=Rows,width=1,n_start=0
},
'n_start=1,width=2' =>
{ Y_axis => 'A005408', # odd 2n+1
# OEIS-Other: A005408 planepath=Rows,width=2 line_type=Y_axis
},
'n_start=1,width=3' =>
{ Y_axis => 'A016777', # 3n+1
# OEIS-Catalogue: A016777 planepath=Rows,width=3 line_type=Y_axis
},
'n_start=1,width=4' =>
{ Y_axis => 'A016813', # 4n+1
# OEIS-Catalogue: A016813 planepath=Rows,width=4 line_type=Y_axis
},
'n_start=1,width=5' =>
{ Y_axis => 'A016861', # 5n+1
# OEIS-Catalogue: A016861 planepath=Rows,width=5 line_type=Y_axis
},
'n_start=1,width=6' =>
{ Y_axis => 'A016921', # 6n+1
# OEIS-Catalogue: A016921 planepath=Rows,width=6 line_type=Y_axis
},
'n_start=1,width=7' =>
{ Y_axis => 'A016993', # 7n+1
# OEIS-Catalogue: A016993 planepath=Rows,width=7 line_type=Y_axis
},
};
}
{ package Math::PlanePath::Columns;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_X_neg_min => undef; # negatives
use constant _NumSeq_X_neg_max => 1; # negatives
sub _NumSeq_Diagonal_NW_min {
my ($self) = @_;
# secret negatives
return ($self->{'height'}-2)*$self->{'height'};
}
use constant _NumSeq_N_oeis_anum =>
{
'n_start=0,height=1' =>
{ X_axis => 'A001477', # integers 0,1,2,3,etc
# OEIS-Other: A001477 planepath=Columns,height=1,n_start=0
},
'n_start=1,height=2' =>
{ X_axis => 'A005408', # odd 2n+1
# OEIS-Other: A005408 planepath=Columns,height=2
},
'n_start=1,height=3' =>
{ X_axis => 'A016777', # 3n+1
# OEIS-Other: A016777 planepath=Columns,height=3
},
'n_start=1,height=4' =>
{ X_axis => 'A016813', # 4n+1
# OEIS-Other: A016813 planepath=Columns,height=4
},
'n_start=1,height=5' =>
{ X_axis => 'A016861', # 5n+1
# OEIS-Other: A016861 planepath=Columns,height=5
},
'n_start=1,height=6' =>
{ X_axis => 'A016921', # 6n+1
# OEIS-Other: A016921 planepath=Columns,height=6
},
'n_start=1,height=7' =>
{ X_axis => 'A016993', # 7n+1
# OEIS-Other: A016993 planepath=Columns,height=7
},
};
}
{ package Math::PlanePath::Diagonals;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'direction=down,n_start=1,x_start=0,y_start=0' =>
{
# Diagonals X_axis -- triangular 1,3,6,etc, but starting i=0 value=1
Y_axis => 'A000124', # triangular+1 = n*(n+1)/2+1
Diagonal => 'A001844', # centred squares 2n(n+1)+1
# OEIS-Catalogue: A000124 planepath=Diagonals line_type=Y_axis
# OEIS-Catalogue: A001844 planepath=Diagonals line_type=Diagonal
},
'direction=up,n_start=1,x_start=0,y_start=0' =>
{
X_axis => 'A000124', # triangular+1 = n*(n+1)/2+1
Diagonal => 'A001844', # centred squares 2n(n+1)+1
# OEIS-Other: A000124 planepath=Diagonals,direction=up
# OEIS-Other: A001844 planepath=Diagonals,direction=up line_type=Diagonal
},
'direction=down,n_start=0,x_start=0,y_start=0' =>
{
X_axis => 'A000096', # n*(n+3)/2
Y_axis => 'A000217', # triangular n*(n+1)/2
# OEIS-Other: A000096 planepath=Diagonals,n_start=0
# OEIS-Other: A000217 planepath=Diagonals,n_start=0 line_type=Y_axis
},
'direction=up,n_start=0,x_start=0,y_start=0' =>
{
X_axis => 'A000217', # triangular n*(n+1)/2
Y_axis => 'A000096', # n*(n+3)/2
# OEIS-Other: A000217 planepath=Diagonals,direction=up,n_start=0
# OEIS-Other: A000096 planepath=Diagonals,direction=up,n_start=0 line_type=Y_axis
},
};
}
{ package Math::PlanePath::DiagonalsAlternating;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ Diagonal => 'A001844', # centred squares 2n(n+1)+1
# OEIS-Other: A001844 planepath=DiagonalsAlternating line_type=Diagonal
# Not quite, extra initial 1 or 0
# X_axis => 'A128918',
# Y_axis => 'A131179',
},
'n_start=0' =>
{ Diagonal => 'A046092', # 2*triangular
# OEIS-Other: A046092 planepath=DiagonalsAlternating,n_start=0 line_type=Diagonal
},
};
}
{ package Math::PlanePath::DiagonalsOctant;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
# 'direction=down,n_start=1' =>
# {
# Not quite, starting i=0 for square=1 cf A000290 starts 0
# # Diagonal => 'A000290', # squares
#
# Not quite, A033638 extra initial 1
# # Y_axis => 'A033638', # quarter squares + 1, 1,1,2,3,5,7,10,13
# }
# 'direction=up,n_start=1' =>
# {
# # Not quite, A002061 extra initial 1
# # Diagonal => 'A002061', # 1,1,3,7,13,21,31,43
# }
'direction=down,n_start=0' =>
{ Diagonal => 'A005563', # n*(n+2) 0,3,8,15,24
# OEIS-Other: A005563 planepath=DiagonalsOctant,n_start=0 line_type=Diagonal
# # Not quite, extra initial 0
# # Y_axis => 'A002620', # quarter squares 0,0,1,2,4,6,9,12,
},
'direction=up,n_start=0' =>
{ Diagonal => 'A002378', # pronic 0,2,6,12,20
# OEIS-Other: A002378 planepath=DiagonalsOctant,direction=up,n_start=0 line_type=Diagonal
# # Not quite, starts n=1 value=0 whereas start Y=0 value=0 here
# # Y_axis => 'A024206', # 0,1,3,5,8,11,15
},
};
}
{ package Math::PlanePath::MPeaks;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 0;
use constant _NumSeq_X_neg_increasing_from_i => 1;
use constant _NumSeq_X_neg_min => 1; # at X=-1,Y=0 rather than X=0,Y=0
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing_from_i => 1;
use constant _NumSeq_Diagonal_NW_min => 2; # at X=-1,Y=1
# MPeaks -- X_axis A045944 matchstick n(3n+2) but initial N=3
# MPeaks -- Diagonal,Y_axis hexagonal first,second spoke, but starting
# from 3
}
{ package Math::PlanePath::Staircase;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
'n_start=1' =>
{ Diagonal => 'A084849',
# OEIS-Other: A084849 planepath=Staircase line_type=Diagonal
},
'n_start=0' =>
{ Diagonal => 'A014105', # second hexagonals
# OEIS-Other: A014105 planepath=Staircase,n_start=0 line_type=Diagonal
},
'n_start=2' =>
{ Diagonal => 'A096376',
# OEIS-Catalogue: A096376 planepath=Staircase,n_start=2 line_type=Diagonal
# Not quite, A128918 has extra initial 1,1
# X_axis => 'A128918',
},
};
}
{ package Math::PlanePath::StaircaseAlternating;
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
return ($self->{'end_type'} eq 'square'
? 1
: 0); # backs-up
}
*_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
'end_type=jump,n_start=1' =>
{ Diagonal => 'A084849',
# OEIS-Other: A084849 planepath=StaircaseAlternating line_type=Diagonal
},
'end_type=jump,n_start=0' =>
{ Diagonal => 'A014105', # second hexagonals
# OEIS-Other: A014105 planepath=StaircaseAlternating,n_start=0 line_type=Diagonal
},
'end_type=jump,n_start=2' =>
{ Diagonal => 'A096376',
# OEIS-Other: A096376 planepath=StaircaseAlternating,n_start=2 line_type=Diagonal
},
'end_type=square,n_start=1' =>
{ Diagonal => 'A058331',
# OEIS-Other: A058331 planepath=StaircaseAlternating,end_type=square line_type=Diagonal
},
'end_type=square,n_start=0' =>
{ Diagonal => 'A001105',
# OEIS-Other: A001105 planepath=StaircaseAlternating,end_type=square,n_start=0 line_type=Diagonal
},
};
}
{ package Math::PlanePath::Corner;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'wider=0,n_start=1' =>
{ Y_axis => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=Corner line_type=Y_axis
},
'wider=0,n_start=0' =>
{ X_axis => 'A005563', # (n+1)^2-1
Y_axis => 'A000290', # squares
Diagonal => 'A002378', # pronic
# OEIS-Other: A005563 planepath=Corner,n_start=0
# OEIS-Other: A000290 planepath=Corner,n_start=0 line_type=Y_axis
# OEIS-Other: A002378 planepath=Corner,n_start=0 line_type=Diagonal
},
'wider=0,n_start=2' =>
{ Y_axis => 'A059100', # n^2+2
Diagonal => 'A014206', # pronic+2
# OEIS-Catalogue: A059100 planepath=Corner,n_start=2 line_type=Y_axis
# OEIS-Catalogue: A014206 planepath=Corner,n_start=2 line_type=Diagonal
},
'wider=1,n_start=0' =>
{ Y_axis => 'A002378', # pronic
Diagonal => 'A005563', # (n+1)^2-1
# OEIS-Other: A002378 planepath=Corner,wider=1,n_start=0 line_type=Y_axis
# OEIS-Other: A005563 planepath=Corner,wider=1,n_start=0 line_type=Diagonal
},
'wider=1,n_start=2' =>
{ Y_axis => 'A014206', # pronic+2
# OEIS-Other: A014206 planepath=Corner,wider=1,n_start=2 line_type=Y_axis
},
'wider=2,n_start=0' =>
{ Y_axis => 'A005563', # (n+1)^2-1
Diagonal => 'A028552', # n(n+3)
# OEIS-Other: A005563 planepath=Corner,wider=2,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A028552 planepath=Corner,wider=2,n_start=0 line_type=Diagonal
},
'wider=2,n_start=1' =>
{ Diagonal => 'A028387', # n(n+3)+1 = n + (n+1)^2
# OEIS-Catalogue: A028387 planepath=Corner,wider=2,n_start=1 line_type=Diagonal
},
'wider=3,n_start=0' =>
{ Y_axis => 'A028552', # n(n+3)
# OEIS-Other: A028552 planepath=Corner,wider=3,n_start=0 line_type=Y_axis
},
};
}
{ package Math::PlanePath::PyramidRows;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # when covered, or single
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
# PyramidRows step=1
do {
my $href =
{ Y_axis => 'A000124', # triangular+1 = n*(n+1)/2+1
};
('step=1,align=centre,n_start=1' => $href,
'step=1,align=right,n_start=1' => $href);
# OEIS-Other: A000124 planepath=PyramidRows,step=1 line_type=Y_axis
# OEIS-Other: A000124 planepath=PyramidRows,step=1,align=right line_type=Y_axis
},
'step=1,align=left,n_start=1' =>
{ Diagonal_NW => 'A000124', # triangular+1 = n*(n+1)/2+1
# OEIS-Other: A000124 planepath=PyramidRows,step=1,align=left line_type=Diagonal_NW
},
do {
my $href =
{ Y_axis => 'A000217', # triangular
};
('step=1,align=centre,n_start=0' => $href,
'step=1,align=right,n_start=0' => $href);
# OEIS-Other: A000217 planepath=PyramidRows,step=1,n_start=0 line_type=Y_axis
# OEIS-Other: A000217 planepath=PyramidRows,step=1,align=right,n_start=0 line_type=Y_axis
},
'step=2,align=centre,n_start=1' =>
{ Diagonal_NW => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=PyramidRows,step=2 line_type=Diagonal_NW
# Not quite, n_start=1 means squares starting from 1 whereas A000290
# starts from 0
# Diagonal => 'A000290',
},
'step=2,align=centre,n_start=0' =>
{ Y_axis => 'A002378', # pronic
Diagonal => 'A005563',
Diagonal_NW => 'A000290', # squares
# OEIS-Other: A002378 planepath=PyramidRows,step=2,n_start=0 line_type=Y_axis
# OEIS-Other: A005563 planepath=PyramidRows,step=2,n_start=0 line_type=Diagonal
# OEIS-Other: A000290 planepath=PyramidRows,step=2,n_start=0 line_type=Diagonal_NW
},
'step=2,align=right,n_start=0' =>
{ Y_axis => 'A000290', # squares
Diagonal => 'A002378', # pronic
# OEIS-Other: A000290 planepath=PyramidRows,step=2,align=right,n_start=0 line_type=Y_axis
# OEIS-Other: A002378 planepath=PyramidRows,step=2,align=right,n_start=0 line_type=Diagonal
},
'step=2,align=left,n_start=0' =>
{ Y_axis => 'A005563',
Diagonal_NW => 'A002378', # pronic
# OEIS-Other: A005563 planepath=PyramidRows,step=2,align=left,n_start=0 line_type=Y_axis
# OEIS-Other: A002378 planepath=PyramidRows,step=2,align=left,n_start=0 line_type=Diagonal_NW
},
'step=2,align=centre,n_start=2' =>
{ Diagonal_NW => 'A059100', # n^2+2
# OEIS-Other: A059100 planepath=PyramidRows,step=2,n_start=2 line_type=Diagonal_NW
},
'step=2,align=right,n_start=2' =>
{ Y_axis => 'A059100', # n^2+2
# OEIS-Other: A059100 planepath=PyramidRows,step=2,align=right,n_start=2 line_type=Y_axis
},
'step=3,align=centre,n_start=1' =>
{ Y_axis => 'A104249',
Diagonal_NW => 'A143689',
# OEIS-Catalogue: A104249 planepath=PyramidRows,step=3 line_type=Y_axis
# OEIS-Catalogue: A143689 planepath=PyramidRows,step=3 line_type=Diagonal_NW
# Not quite OFFSET=1 cf start i=0 here
# Diagonal => 'A005448',
# # OEIS-Catalogue: A005448 planepath=PyramidRows,step=3 line_type=Diagonal
},
'step=3,align=right,n_start=1' =>
{ Y_axis => 'A143689',
Diagonal => 'A104249',
# OEIS-Other: A143689 planepath=PyramidRows,step=3,align=right line_type=Y_axis
# OEIS-Other: A104249 planepath=PyramidRows,step=3,align=right line_type=Diagonal
# Not quite OFFSET=1 cf start i=0 here
# Diagonal => 'A005448',
# # OEIS-Catalogue: A005448 planepath=PyramidRows,step=3 line_type=Diagonal
},
'step=3,align=centre,n_start=0' =>
{ Y_axis => 'A005449', # second pentagonal n*(3n+1)/2
Diagonal_NW => 'A000326', # pentagonal n(3n-1)/2
# OEIS-Other: A005449 planepath=PyramidRows,step=3,n_start=0 line_type=Y_axis
# OEIS-Other: A000326 planepath=PyramidRows,step=3,n_start=0 line_type=Diagonal_NW
},
'step=3,align=right,n_start=0' =>
{ Y_axis => 'A000326', # pentagonal n(3n-1)/2
Diagonal => 'A005449', # second pentagonal n*(3n+1)/2
# OEIS-Other: A000326 planepath=PyramidRows,step=3,align=right,n_start=0 line_type=Y_axis
# OEIS-Other: A005449 planepath=PyramidRows,step=3,align=right,n_start=0 line_type=Diagonal
},
'step=4,align=centre,n_start=1' =>
{ Y_axis => 'A084849',
Diagonal => 'A001844',
Diagonal_NW => 'A058331',
# OEIS-Catalogue: A084849 planepath=PyramidRows,step=4 line_type=Y_axis
# OEIS-Other: A001844 planepath=PyramidRows,step=4 line_type=Diagonal
# OEIS-Other: A058331 planepath=PyramidRows,step=4 line_type=Diagonal_NW
},
'step=4,align=right,n_start=1' =>
{ Diagonal => 'A058331',
# OEIS-Other: A058331 planepath=PyramidRows,step=4,align=right line_type=Diagonal
},
'step=4,align=left,n_start=1' =>
{ Diagonal_NW => 'A001844',
# OEIS-Other: A001844 planepath=PyramidRows,step=4,align=left line_type=Diagonal_NW
},
'step=4,align=centre,n_start=0' =>
{ Y_axis => 'A014105', # second hexagonal
Diagonal => 'A046092', # 4*triangular
Diagonal_NW => 'A001105', # 2*n^2
# OEIS-Other: A014105 planepath=PyramidRows,step=4,n_start=0 line_type=Y_axis
# OEIS-Catalogue: A046092 planepath=PyramidRows,step=4,n_start=0 line_type=Diagonal
# OEIS-Other: A001105 planepath=PyramidRows,step=4,n_start=0 line_type=Diagonal_NW
},
'step=4,align=right,n_start=0' =>
{ Y_axis => 'A000384', # 2*n^2-n, hexagonal numbers
Diagonal => 'A001105',
# OEIS-Other: A000384 planepath=PyramidRows,step=4,align=right,n_start=0 line_type=Y_axis
# OEIS-Other: A001105 planepath=PyramidRows,step=4,align=right,n_start=0 line_type=Diagonal
},
'step=4,align=left,n_start=0' =>
{ Diagonal_NW => 'A046092', # 4*triangular
# OEIS-Other: A046092 planepath=PyramidRows,step=4,align=left,n_start=0 line_type=Diagonal_NW
},
# TODO PyramidRows,step=5 n_start=0
'step=5,align=centre,n_start=1' =>
{ Y_axis => 'A116668',
# OEIS-Other: A116668 planepath=PyramidRows,step=5 line_type=Y_axis
},
'step=6,align=centre,n_start=1' =>
{ Diagonal_NW => 'A056107',
Y_axis => 'A056108',
Diagonal => 'A056109',
# OEIS-Other: A056107 planepath=PyramidRows,step=6 line_type=Diagonal_NW
# OEIS-Other: A056108 planepath=PyramidRows,step=6 line_type=Y_axis
# OEIS-Other: A056109 planepath=PyramidRows,step=6 line_type=Diagonal
},
'step=8,align=centre,n_start=1' =>
{ Diagonal_NW => 'A053755',
# OEIS-Other: A053755 planepath=PyramidRows,step=8 line_type=Diagonal_NW
},
'step=9,align=centre,n_start=1' =>
{ Y_axis => 'A006137',
Diagonal => 'A038764',
# OEIS-Other: A006137 planepath=PyramidRows,step=9 line_type=Y_axis
# OEIS-Other: A038764 planepath=PyramidRows,step=9 line_type=Diagonal
},
};
}
{ package Math::PlanePath::PyramidSides;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ X_neg => 'A002522',
Diagonal => 'A033951',
# OEIS-Catalogue: A002522 planepath=PyramidSides line_type=X_neg
# OEIS-Other: A033951 planepath=PyramidSides line_type=Diagonal
#
# X_axis -- squares (x+1)^2, but starting i=0 value=1
} };
}
{ package Math::PlanePath::CellularRule;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'rule=5,n_start=1' =>
{ Y_axis => 'A061925', # ceil(n^2/2)+1
# OEIS-Catalogue: A061925 planepath=CellularRule,rule=5 line_type=Y_axis
},
# rule 84,116,212,244 two-wide right line
do {
my $tworight
= { Diagonal => 'A005408', # odds 2n+1
};
('rule=84,n_start=1' => $tworight,
'rule=116,n_start=1' => $tworight,
'rule=212,n_start=1' => $tworight,
'rule=244,n_start=1' => $tworight,
);
# OEIS-Other: A005408 planepath=CellularRule,rule=84 line_type=Diagonal
# OEIS-Other: A005408 planepath=CellularRule,rule=116 line_type=Diagonal
# OEIS-Other: A005408 planepath=CellularRule,rule=212 line_type=Diagonal
# OEIS-Other: A005408 planepath=CellularRule,rule=244 line_type=Diagonal
},
'rule=77,n_start=1' =>
{ Y_axis => 'A000124', # triangular+1
# OEIS-Other: A000124 planepath=CellularRule,rule=77 line_type=Y_axis
},
'rule=177,n_start=1' =>
{ Diagonal => 'A000124', # triangular+1
# OEIS-Other: A000124 planepath=CellularRule,rule=177 line_type=Diagonal
},
'rule=185,n_start=1' =>
{ Diagonal => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=CellularRule,rule=185 line_type=Diagonal
},
'rule=189,n_start=1' =>
{ Y_axis => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=CellularRule,rule=189 line_type=Y_axis
},
# PyramidRows step=1,align=left
# OEIS-Other: A000124 planepath=CellularRule,rule=206 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=238 line_type=Diagonal_NW
do {
my $solidgapright
= { Diagonal => 'A002522', # n^2+1
};
('rule=209,n_start=1' => $solidgapright,
'rule=241,n_start=1' => $solidgapright,
);
# OEIS-Other: A002522 planepath=CellularRule,rule=209 line_type=Diagonal
# OEIS-Other: A002522 planepath=CellularRule,rule=241 line_type=Diagonal
},
'rule=29,n_start=1' =>
{ Y_axis => 'A000124', # triangular+1
# OEIS-Other: A000124 planepath=CellularRule,rule=29 line_type=Y_axis
},
'rule=221,n_start=1' =>
{ Y_axis => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=CellularRule,rule=221 line_type=Y_axis
},
'rule=229,n_start=1' =>
{ Y_axis => 'A002522', # n^2+1
# OEIS-Other: A002522 planepath=CellularRule,rule=229 line_type=Y_axis
},
#
# rule=13 Y axis
#
# rule=28,156
# Y_axis => 'A002620', quarter squares floor(n^2/4) but diff start
# Diagonal => 'A024206', quarter squares - 1, but diff start
#
# A000027 naturals integers 1 upwards, but OFFSET=1 cf start Y=0 here
# # central column only
# 'rule=4' =>
# { Y_axis => 'A000027', # 1 upwards
# # OEIS-Other: A000027 planepath=CellularRule,rule=4 line_type=Y_axis
# },
#
# # right line only rule=16,24,48,56,80,88,112,120,144,152,176,184,208,216,240,248
# Not quite A000027 OFFSET=1 vs start X=Y=0 here
# 'rule=16' =>
# { Y_axis => 'A000027', # 1 upwards
# # OEIS-Other: A000027 planepath=CellularRule,rule=16 line_type=Diagonal
# },
};
}
{
package Math::PlanePath::CellularRule::Line;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
}
{
package Math::PlanePath::CellularRule::OneTwo;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
# use constant _NumSeq_N_oeis_anum =>
# { 'align=left,n_start=1' =>
# {
# Not quite, OFFSET=1 cf coordinate X=0 here
# Diagonal_NW => 'A001651', # not divisible by 3
# },
# { 'align=right,n_start=1' =>
# Not quite, A032766 0 or 1 mod 3, but it starts OFFSET=0 value=0
# whereas path start 1,3,4,etc without initial 0
# Diagonal => 'A032766',
#
# };
}
{
package Math::PlanePath::CellularRule::Two;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'align=right,n_start=1' =>
{ Diagonal => 'A005408', # odd 2n+1
# OEIS-Other: A005408 planepath=CellularRule,rule=84 line_type=Diagonal
},
};
}
{
package Math::PlanePath::CellularRule::OddSolid;
# rule=50,58,114,122,178,179,186,242,250 pyramid every second point
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ Diagonal_NW => 'A000124', # triangular+1
# OEIS-Other: A000124 planepath=CellularRule,rule=50 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=58 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=114 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=122 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=178 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=179 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=186 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=242 line_type=Diagonal_NW
# OEIS-Other: A000124 planepath=CellularRule,rule=250 line_type=Diagonal_NW
#
# Not quite, starts value=0
# Diagonal => 'A000217', # triangular numbers but diff start
},
};
}
{ package Math::PlanePath::CellularRule54;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
}
{ package Math::PlanePath::CellularRule57;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
}
{ package Math::PlanePath::CellularRule190;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'mirror=0,n_start=0' =>
{ Diagonal_NW => 'A006578', # triangular and quarter square
# OEIS-Catalogue: A006578 planepath=CellularRule190,n_start=0 line_type=Diagonal_NW
},
'mirror=1,n_start=0' =>
{ Diagonal_NW => 'A006578', # triangular and quarter square
# OEIS-Other: A006578 planepath=CellularRule190,mirror=1,n_start=0 line_type=Diagonal_NW
},
};
}
{ package Math::PlanePath::UlamWarburton;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'parts=4,n_start=0' =>
{ Depth_start => 'A147562', # cells ON after n stages
# OEIS-Catalogue: A147562 planepath=UlamWarburton,n_start=0 line_type=Depth_start
},
'parts=2,n_start=0' =>
{ Depth_start => 'A183060', # num cells ON, starting from 0
X_axis => 'A183060', # X_axis == Depth_start
# OEIS-Catalogue: A183060 planepath=UlamWarburton,parts=2,n_start=0 line_type=Depth_start
# OEIS-Other: A183060 planepath=UlamWarburton,parts=2,n_start=0
},
'parts=1,n_start=1' =>
{ Depth_end => 'A151922', # num cells ON at end of depth=n
Y_axis => 'A151922', # Y_axis == Depth_end
# OEIS-Catalogue: A151922 planepath=UlamWarburton,parts=1,n_start=1 line_type=Depth_end
# OEIS-Other: A151922 planepath=UlamWarburton,parts=1,n_start=1 line_type=Y_axis
},
};
}
{ package Math::PlanePath::UlamWarburtonQuarter;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
# low 10111=23 increment to 11000=24
# 4^2*3+4*3^2+1*3^3 = 111 123
# 4^3*3 = 192 150 +27 = 3^3
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ Depth_end => 'A151920', # 3^count1bits(n), OFFSET=0 1,2,5,6,9
# OEIS-Catalogue: A151920 planepath=UlamWarburtonQuarter line_type=Depth_end
},
};
}
{ package Math::PlanePath::CoprimeColumns;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_X_offset => 1;
# CoprimeColumns
# X_axis => 'A002088', # cumulative totient but start X=1 value=0;
# Diagonal A015614 cumulative-1 but start X=1 value=1
}
{ package Math::PlanePath::DivisibleColumns;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
# Not quite, X_axis => 'A006218' but path start X=1 cf OFFSET=0,
# Not quite, Diagonal => 'A077597' but path start X=1 cf OFFSET=0
}
# { package Math::PlanePath::File;
# # File points from a disk file
# # FIXME: analyze points for min/max
# }
# { package Math::PlanePath::QuintetCurve;
# }
# { package Math::PlanePath::QuintetCentres;
# # inherit QuintetCurve
# }
{ package Math::PlanePath::CornerReplicate;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ '' =>
{ X_axis => 'A000695', # base 4 digits 0,1 only
Y_axis => 'A001196', # base 4 digits 0,3 only
Diagonal => 'A062880', # base 4 digits 0,2 only
# OEIS-Other: A000695 planepath=CornerReplicate
# OEIS-Other: A001196 planepath=CornerReplicate line_type=Y_axis
# OEIS-Other: A062880 planepath=CornerReplicate line_type=Diagonal
},
};
}
{ package Math::PlanePath::DigitGroups;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'radix=2,i_start=1' =>
{ X_axis => 'A084471', # 0 -> 00 in binary, starting OFFSET=1
# OEIS-Catalogue: A084471 planepath=DigitGroups,radix=2 i_start=1
},
};
}
{ package Math::PlanePath::FibonacciWordFractal;
use constant _NumSeq_X_axis_increasing => 1; # when touched
use constant _NumSeq_Y_axis_increasing => 1; # when touched
use constant _NumSeq_Diagonal_increasing => 1; # when touched
}
{ package Math::PlanePath::LTiling;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{ 'L_fill=middle' =>
{ Diagonal => 'A062880', # base 4 digits 0,2 only
# OEIS-Other: A062880 planepath=LTiling line_type=Diagonal
},
};
}
{ package Math::PlanePath::WythoffArray;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_N_oeis_anum =>
{
'x_start=1,y_start=1' =>
{ Y_axis => 'A003622', # spectrum of phi 1,4,6,9
Diagonal => 'A020941', # diagonal, OFFSET=1
# OEIS-Catalogue: A003622 planepath=WythoffArray,x_start=1,y_start=1 line_type=Y_axis
# OEIS-Catalogue: A020941 planepath=WythoffArray,x_start=1,y_start=1 line_type=Diagonal
# Y=1 every second => 'A005248', # every second Lucas number
},
# # Not quite, extra initial 0,1 in A000045 Fibonaccis
# # X_axis => 'A000045',
#
# # Y=1 row X=0,2,4,etc => 'A005248', # every second Lucas number
};
}
{ package Math::PlanePath::WythoffPreliminaryTriangle;
use constant _NumSeq_Y_axis_i_start => 1;
use constant _NumSeq_N_oeis_anum =>
{
'' =>
{ Y_axis => 'A173027',
# OEIS-Other: A173027 planepath=WythoffPreliminaryTriangle line_type=Y_axis
},
};
}
{ package Math::PlanePath::PowerArray;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
# cf Not quite A168183 non-multiples-of-9, A168186 non-multiples-of-12
# are values on Y axis, except OFFSET=1 value=1, whereas path start Y=0
# value=1
use constant _NumSeq_N_oeis_anum =>
{ 'radix=2' =>
{ X_axis => 'A000079', # powers 2^X
Y_axis => 'A005408', # odd 2n+1
Diagonal => 'A014480', # (2n+1)*2^n starting n=0
# OEIS-Other: A000079 planepath=PowerArray
# OEIS-Other: A005408 planepath=PowerArray line_type=Y_axis
# OEIS-Catalogue: A014480 planepath=PowerArray line_type=Diagonal
},
'radix=3' =>
{ X_axis => 'A000244', # powers 3^X
# OEIS-Other: A000244 planepath=PowerArray,radix=3
#
# Not quite, OFFSET=1 cf path start Y=0
# Y_axis => 'A001651', # non multiples of 3
},
'radix=4' =>
{ X_axis => 'A000302', # powers 4^X
# OEIS-Other: A000302 planepath=PowerArray,radix=4
},
'radix=5' =>
{ X_axis => 'A000351', # powers 5^X
# OEIS-Other: A000351 planepath=PowerArray,radix=5
},
'radix=10' =>
{ X_axis => 'A011557', # powers 10^X
# OEIS-Other: A011557 planepath=PowerArray,radix=10
# Not quite, A067251 OFFSET=1 value=1 whereas path Y=0 N=value=1
# Y_axis => 'A067251', # no trailing 0 digits
# # OEIS-Catalogue: A067251 planepath=PowerArray,radix=10 line_type=Y_axis
},
};
}
#------------------------------------------------------------------------------
# Math-PlanePath-Toothpick
{ package Math::PlanePath::ToothpickTree;
# X axis has N=0 or N=0,1 only, except for parts=octant axis at Y=1
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
return ($self->{'parts'} ne 'octant');
}
# Y axis has N=0,N=1 only, except for parts=octant_up axis at X=1
sub _NumSeq_Y_axis_increasing {
my ($self) = @_;
return ($self->{'parts'} ne 'octant_up');
}
use constant _NumSeq_Y_neg_increasing => 1; # N=0,N=2 only
use constant _NumSeq_Diagonal_increasing => 1; # growth steps along diags
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ 'parts=4' =>
{ Depth_start => 'A139250',
# OEIS-Other: A139250 planepath=ToothpickTree,parts=4 line_type=Depth_start
},
'parts=3' =>
{ Depth_start => 'A153006',
# OEIS-Other: A153006 planepath=ToothpickTree,parts=3 line_type=Depth_start
},
'parts=2' =>
{ Depth_start => 'A152998',
# OEIS-Other: A152998 planepath=ToothpickTree,parts=2 line_type=Depth_start
},
'parts=1' =>
{ Depth_start => 'A153000',
# OEIS-Other: A153000 planepath=ToothpickTree,parts=1 line_type=Depth_start
},
'parts=wedge' =>
{ Depth_start => 'A160406',
# OEIS-Other: A160406 planepath=ToothpickTree,parts=wedge line_type=Depth_start
},
'parts=two_horiz' =>
{ Depth_start => 'A160158',
# OEIS-Other: A160158 planepath=ToothpickTree,parts=two_horiz line_type=Depth_start
},
};
}
{ package Math::PlanePath::ToothpickReplicate;
use constant _NumSeq_Y_axis_increasing => 1; # N=0,N=1 only
sub _NumSeq_Y_neg_increasing {
my ($self) = @_;
return ($self->{'parts'} == 3 ? 0 # replication twist
: 1); # N=0,N=2 only
}
use constant _NumSeq_Diagonal_increasing => 1; # replicate along diags
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
}
{ package Math::PlanePath::ToothpickUpist;
use constant _NumSeq_Y_axis_increasing => 1; # rows increasing
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ '' =>
{ Depth_start => 'A151566',
# OEIS-Other: A151566 planepath=ToothpickUpist line_type=Depth_start
},
};
}
{ package Math::PlanePath::ToothpickSpiral;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ 'n_start=1' =>
{ Diagonal => 'A014634', # odd-index hexagonals
Diagonal_NW => 'A033567', #
Diagonal_SW => 'A185438', #
Diagonal_SE => 'A188135', #
# OEIS-Other: A014634 planepath=ToothpickSpiral line_type=Diagonal
# OEIS-Other: A033567 planepath=ToothpickSpiral line_type=Diagonal_NW
# OEIS-Other: A185438 planepath=ToothpickSpiral line_type=Diagonal_SW
# OEIS-Other: A188135 planepath=ToothpickSpiral line_type=Diagonal_SE
},
'n_start=0' =>
{ Diagonal => 'A033587', #
Diagonal_SW => 'A014635', # even-index hexagonals
Diagonal_SE => 'A033585', #
# OEIS-Other: A033587 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal
# OEIS-Other: A014635 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal_SW
# OEIS-Other: A033585 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal_SE
},
};
}
{ package Math::PlanePath::LCornerTree;
sub _NumSeq_X_axis_increasing {
my ($self) = @_;
return $self->{'parts'} ne 'diagonal-1';
}
{
my %_NumSeq_Y_axis_increasing = ('octant+1' => 1, # two points only
'diagonal-1' => 1,
);
sub _NumSeq_Y_axis_increasing {
my ($self) = @_;
return $_NumSeq_Y_axis_increasing{$self->{'parts'}};
}
}
sub _NumSeq_X_neg_increasing {
my ($self) = @_;
return ($self->{'parts'} eq 'wedge' # two points N=0,N=1
|| $self->{'parts'} eq 'wedge+1' # three points N=0,1,7
|| $self->{'parts'} eq 'diagonal');
}
# parts=diagonal has minimum N=0 at X=0,Y=-1, so explicit Y_neg minimum
use constant _NumSeq_Y_neg_min => 0;
sub _NumSeq_Y_neg_increasing {
my ($self) = @_;
return $self->{'parts'} ne 'diagonal';
}
use constant _NumSeq_Diagonal_increasing => 1; # growth along diags
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ 'parts=4' =>
{ Depth_start => 'A160410', # 4 * cumulative 3^count1bits(n)
# OEIS-Other: A160410 planepath=LCornerTree line_type=Depth_start
},
'parts=3' =>
{ Depth_start => 'A160412', # 3 * cumulative 3^count1bits(n)
# OEIS-Other: A160412 planepath=LCornerTree,parts=3 line_type=Depth_start
},
'parts=diagonal-1' =>
{ Depth_start => 'A183148', # half-plane triplet toothpicks
# OEIS-Other: A183148 planepath=LCornerTree,parts=diagonal-1 line_type=Depth_start
# No, N=1 start of depth=1 is at X=1,Y=0 not on SE diagonal
# Diagonal_SE => 'A183148',
},
# Not quite, A130665=1,4,7,16 offset=0 whereas Nend=0,1,4,7,16 depth=0
# has extra initial 0.
# 'parts=1' =>
# { Depth_end => 'A130665', # cumulative 3^count1bits(d), starting a(0)=1
# # OEIS-Catalogue: A130665 planepath=LCornerTree,parts=1,n_start=-1 line_type=Depth_end i_offset=1
# },
};
}
{ package Math::PlanePath::LCornerReplicate;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1; # replicate along diags
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ '' =>
{ Diagonal => 'A062880', # base 4 digits 0,2 only
# OEIS-Other: A062880 planepath=LCornerReplicate line_type=Diagonal
},
};
}
{ package Math::PlanePath::OneOfEight;
use constant _NumSeq_X_axis_increasing => 1;
use constant _NumSeq_X_neg_increasing => 1;
use constant _NumSeq_Y_axis_increasing => 1;
use constant _NumSeq_Y_neg_increasing => 1;
use constant _NumSeq_Diagonal_increasing => 1;
use constant _NumSeq_Diagonal_NW_increasing => 1;
use constant _NumSeq_Diagonal_SW_increasing => 1;
use constant _NumSeq_Diagonal_SE_increasing => 1;
# catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick
use constant _NumSeq_N_oeis_anum =>
{ 'parts=4' =>
{ Depth_start => 'A151725',
# OEIS-Other: A151725 planepath=OneOfEight line_type=Depth_start
},
'parts=1' =>
{ Depth_start => 'A151735',
# OEIS-Other: A151735 planepath=OneOfEight,parts=1 line_type=Depth_start
},
'parts=3mid' =>
{ Depth_start => 'A170880',
# OEIS-Other: A170880 planepath=OneOfEight,parts=3mid line_type=Depth_start
},
'parts=3side' =>
{ Depth_start => 'A170879',
# OEIS-Other: A170879 planepath=OneOfEight,parts=3side line_type=Depth_start
},
};
}
{ package Math::PlanePath::HTree;
# clockwise around each sub-tree so N increases along X axis
use constant _NumSeq_X_axis_increasing => 1;
}
#------------------------------------------------------------------------------
{ package Math::PlanePath;
use constant _NumSeq_A2 => 0;
}
{ package Math::PlanePath::TriangleSpiral;
use constant _NumSeq_A2 => 1;
}
{ package Math::PlanePath::HexSpiral;
use constant _NumSeq_A2 => 1;
}
{ package Math::PlanePath::HexArms;
use constant _NumSeq_A2 => 1;
}
{ package Math::PlanePath::TriangularHypot;
use constant _NumSeq_A2 => 1;
}
{ package Math::PlanePath::Flowsnake;
use constant _NumSeq_A2 => 1;
# and FlowsnakeCentres inherits
}
1;
__END__
=for stopwords Ryde Math-PlanePath SquareSpiral lookup PlanePath ie
=head1 NAME
Math::NumSeq::PlanePathN -- sequence of N values from PlanePath module
=head1 SYNOPSIS
use Math::NumSeq::PlanePathN;
my $seq = Math::NumSeq::PlanePathN->new (planepath => 'SquareSpiral',
line_type => 'X_axis');
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
This module presents N values from a C<Math::PlanePath> as a sequence. The
default is the X axis, or the C<line_type> parameter (a string) can choose
among
"X_axis" X axis
"Y_axis" Y axis
"X_neg" X negative axis
"Y_neg" Y negative axis
"Diagonal" leading diagonal X=i, Y=i
"Diagonal_NW" north-west diagonal X=-i, Y=i
"Diagonal_SW" south-west diagonal X=-i, Y=-i
"Diagonal_SE" south-east diagonal X=i, Y=-i
"Depth_start" first N at depth=i
"Depth_end" last N at depth=i
For example the C<SquareSpiral> X axis starts i=0 with values 1, 2, 11, 28,
53, 86, etc.
"X_neg", "Y_neg", "Diagonal_NW", etc, on paths which don't traverse negative
X or Y have just a single value from X=0,Y=0.
The behaviour on paths which visit only some of the points on the respective
axis is unspecified as yet, as is behaviour on paths with repeat points,
such as the C<DragonCurve>.
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::PlanePathN-E<gt>new (key=E<gt>value,...)>
Create and return a new sequence object. The options are
planepath string, name of a PlanePath module
planepath_object PlanePath object
line_type string, as described above
C<planepath> can be either the module part such as "SquareSpiral" or a
full class name "Math::PlanePath::SquareSpiral".
=item C<$value = $seq-E<gt>ith($i)>
Return the N value at C<$i> in the PlanePath. C<$i> gives a position on the
respective C<line_type>, so the X,Y to lookup a C<$value=N> is
X,Y line_type
----- ---------
$i, 0 "X_axis"
0, $i "Y_axis"
-$i, 0 "X_neg"
0, -$i "Y_neg"
$i, $i "Diagonal"
$i, -$i "Diagonal_NW"
-$i, -$i "Diagonal_SW"
$i, -$i "Diagonal_SE"
=item C<$bool = $seq-E<gt>pred($value)>
Return true if C<$value> occurs in the sequence.
This means C<$value> is an integer N which is on the respective
C<line_type>, ie. that C<($path-E<gt>n_to_xy($value)> is on the line type.
=back
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::PlanePathCoord>,
L<Math::NumSeq::PlanePathDelta>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-planepath/index.html>
=head1 LICENSE
Copyright 2011, 2012, 2013, 2014 Kevin Ryde
This file is part of Math-PlanePath.
Math-PlanePath is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the Free
Software Foundation; either version 3, or (at your option) any later
version.
Math-PlanePath is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
more details.
You should have received a copy of the GNU General Public License along with
Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
=cut