# Copyright 2012, 2013, 2014 Kevin Ryde
# This file is part of Math-NumSeq.
#
# Math-NumSeq 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-NumSeq 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-NumSeq. If not, see <http://www.gnu.org/licenses/>.
# period(period(...period(m)))
# p^(w+1)[m]=p^w[m] at w=Fibonacci frequency, for w>=1
# A001178 Fibonacci frequency of n.
#
package Math::NumSeq::PisanoPeriodSteps;
use 5.004;
use strict;
use vars '$VERSION', '@ISA';
$VERSION = 71;
use Math::NumSeq;
use Math::NumSeq::Base::IterateIth;
@ISA = ('Math::NumSeq::Base::IterateIth',
'Math::NumSeq');
*_is_infinite = \&Math::NumSeq::_is_infinite;
use Math::NumSeq::Base::Cache
'cache_hash';
use Math::NumSeq::NumAronson 8; # new in v.8
*_round_down_pow = \&Math::NumSeq::NumAronson::_round_down_pow;
use Math::NumSeq::PisanoPeriod;
# uncomment this to run the ### lines
#use Smart::Comments;
# use constant name => Math::NumSeq::__('...');
use constant i_start => 1;
use constant characteristic_smaller => 1;
use constant characteristic_integer => 1;
use constant values_min => 0;
use constant parameter_info_array =>
[ { name => 'values_type',
display => Math::NumSeq::__('Values Type'),
type => 'enum',
default => 'freq',
choices => ['freq',
'log'],
choices_display => [Math::NumSeq::__('Freq'),
Math::NumSeq::__('Log')],
description => Math::NumSeq::__('The "frequency" count of steps, or the "logarithm" power in the final repeating period.'),
},
];
sub description {
my ($self) = @_;
if (ref $self && $self->{'values_type'} eq 'log') {
return Math::NumSeq::__('Leonardo logarithm, the "l" exponent in the final period 24*5^(l-1) on reaching an unchanging PisanoPeriod.');
} else {
return Math::NumSeq::__('Fibonacci frequency, how many applications of the PisanoPeriod to reach an unchanging value.')
}
}
sub characteristic_count {
my ($self) = @_;
return ($self->{'values_type'} eq 'freq');
}
#------------------------------------------------------------------------------
my %oeis_anum = (freq => 'A001178',
log => 'A001179',
# OEIS-Catalogue: A001178
# OEIS-Catalogue: A001179 values_type=log
);
sub oeis_anum {
my ($self) = @_;
### oeis_anum(): $self
return $oeis_anum{$self->{'values_type'}};
}
#------------------------------------------------------------------------------
sub ith {
my ($self, $i) = @_;
### PisanoPeriodSteps ith(): $i
if (_is_infinite($i)) {
return $i;
}
if ($i <= 1) {
if ($i < 1) {
return undef;
}
return 0;
}
my $key = "PisanoPeriodSteps:" . $self->{'values_type'};
my $value = cache_hash()->{$key.$i};
if (! defined $value) {
### calculate ...
my $count = -1; # default undef for outside range
my $log = -1;
my @pending = ($i);
if (defined ($i = $self->Math::NumSeq::PisanoPeriod::ith($i))) {
for (;;) {
### at: "i=$i"
my $p = $self->Math::NumSeq::PisanoPeriod::ith($i);
if (! defined $p) {
### outside range of PisanoPeriod ...
last;
}
if ($p == $i) {
### same: "i=$i p=$p"
$count = 0;
# $i is the final period, turn it into the logarithm
$i /= 24;
($i, $log) = _round_down_pow ($i, 5);
$log++;
last;
}
### not same: "i=$i p=$p"
if (defined ($count = cache_hash()->{"PisanoPeriodSteps:freq:".$i})) {
$log = cache_hash()->{"PisanoPeriodSteps:log:".$i};
### found cache: "i=$i count=$count log=$log"
last;
}
push @pending, $i;
$i = $p;
}
### @pending
foreach (reverse @pending) {
if ($count >= 0) { $count++; }
### store: "$_ count $count"
cache_hash()->{"PisanoPeriodSteps:freq:".$_} = $count;
cache_hash()->{"PisanoPeriodSteps:log:".$_} = $log;
}
$value = ($self->{'values_type'} eq 'freq' ? $count : $log);
}
}
### return: $value
return ($value >= 0 ? $value : undef);
}
1;
__END__
# sub ith {
# my ($self, $i) = @_;
# ### PisanoPeriodSteps ith(): $i
#
# if (_is_infinite($i)) {
# return $i;
# }
#
# $i = $self->Math::NumSeq::PisanoPeriod::ith($i);
# if (! defined $i) {
# return undef;
# }
# my $count = 1;
#
# for (;;) {
# my $p = $self->Math::NumSeq::PisanoPeriod::ith($i);
# if (! defined $p) {
# return undef;
# }
# if ($p == $i) {
# return $count;
# }
# $i = $p;
# $count++;
# }
# }
=for stopwords Ryde Math-NumSeq ie
=head1 NAME
Math::NumSeq::PisanoPeriodSteps -- Fibonacci frequency and Leonardo logarithm
=head1 SYNOPSIS
use Math::NumSeq::PisanoPeriodSteps;
my $seq = Math::NumSeq::PisanoPeriodSteps->new;
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
This is the number of times the C<PisanoPeriod> must be applied before
reaching an unchanging value.
0, 4, 3, 2, 3, 1, 2, 2, 1, 2, 3, 1, 3, 2, 3, 1, 2, 1, 2, ...
starting i=1
X<Fulton, D.>X<Morris, W.L.>As per Fulton and Morris
=over
"On arithmetical functions related to the Fibonacci numbers",
Acta Arithmetica, volume 16, 1969, pages 105-110.
L<http://matwbn.icm.edu.pl/ksiazki/aa/aa16/aa1621.pdf>
=back
repeatedly applying the PisanoPeriod eventually reaches an m which is
unchanging, ie. for which PisanoPeriod(m)==m. For example i=5 goes
PisanoPeriod(5)=20
PisanoPeriod(20)=60
PisanoPeriod(60)=60
PisanoPeriod(120)=120
so value=3 applications until to reach unchanging 120
=head2 Leonardo Logarithm
The unchanging period reached is always of the form
m = 24 * 5^(l-1)
The "l" exponent is the Leonardo logarithm. Option C<values_type =E<gt>
"log"> returns that as the sequence values.
0, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, ...
starting i=1
For example the i=5 above ends at m=120=24*5^1 so l-1=1 is l=2 for the
sequence value.
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::PisanoPeriodSteps-E<gt>new ()>
=item C<$seq = Math::NumSeq::PisanoPeriodSteps-E<gt>new (values_type =E<gt> $str)>
Create and return a new sequence object.
=back
=head2 Random Access
=over
=item C<$value = $seq-E<gt>ith($i)>
Return the count or logarithm of C<$i>.
=cut
=back
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::Fibonacci>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-numseq/index.html>
=head1 LICENSE
Copyright 2012, 2013, 2014 Kevin Ryde
Math-NumSeq 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-NumSeq 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-NumSeq. If not, see <http://www.gnu.org/licenses/>.
=cut