Kevin Ryde > Math-PlanePath-116 > Math::PlanePath::PyramidRows

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Module Version: 116   Source   Latest Release: Math-PlanePath-117

NAME ^

Math::PlanePath::PyramidRows -- points stacked up in a pyramid

SYNOPSIS ^

 use Math::PlanePath::PyramidRows;
 my $path = Math::PlanePath::PyramidRows->new;
 my ($x, $y) = $path->n_to_xy (123);

DESCRIPTION ^

This path arranges points in successively wider rows going upwards so as to form an upside-down pyramid. The default step is 2, ie. each row 2 wider than the preceding, an extra point at the left and the right,

    17  18  19  20  21  22  23  24  25         4
        10  11  12  13  14  15  16             3
             5   6   7   8   9                 2
                 2   3   4                     1
                     1                   <-  Y=0

    -4  -3  -2  -1  X=0  1   2   3   4 ...

The right end N=1,4,9,16,etc is the perfect squares. The vertical 2,6,12,20,etc at x=-1 is the pronic numbers s*(s+1), half way between those successive squares.

The step 2 is the same as the PyramidSides, Corner and SacksSpiral paths. For the SacksSpiral, spiral arms going to the right correspond to diagonals in the pyramid, and arms to the left correspond to verticals.

Step Parameter

A step parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4

    my $path = Math::PlanePath::PyramidRows->new (step => 4);

makes each row 2 wider on each side successively

   29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45        4
         16 17 18 19 20 21 22 23 24 25 26 27 28              3
                7  8  9 10 11 12 13 14 15                    2
                      2  3  4  5  6                          1
                            1                          <-  Y=0

         -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6 ...

If the step is an odd number then the extra is at the right, so step 3 gives

    13  14  15  16  17  18  19  20  21  22        3
         6   7   8   9  10  11  12                2
             2   3   4   5                        1
                 1                          <-  Y=0

    -3  -2  -1  X=0  1   2   3   4 ...

Or step 1 goes solely to the right. This is equivalent to the Diagonals path, but columns shifted up to make horizontal rows.

    step => 1

    11  12  13  14  15                4
     7   8   9  10                    3
     4   5   6                        2
     2   3                            1
     1                          <-  Y=0

    X=0  1   2   3   4 ...

Step 0 means simply a vertical, each row 1 wide and not increasing. This is unlikely to be much use. The Rows path with width 1 does this too.

    step => 0

     5        4
     4        3
     3        2
     2        1
     1    <-y=0

    X=0

Various number sequences fall in regular patterns positions depending on the step. Large steps are not particularly interesting and quickly become very wide. A limit might be desirable in a user interface, but there's no limit in the code as such.

Align Parameter

An optional align parameter controls how the points are arranged relative to the Y axis. The default shown above is "centre".

"right" means points to the right of the axis,

    align=>"right"

    26  27  28  29  30  31  32  33  34  35  36        5
    17  18  19  20  21  22  23  24  25                4
    10  11  12  13  14  15  16                        3
     5   6   7   8   9                                2
     2   3   4                                        1
     1                                            <- Y=0

    X=0  1   2   3   4   5   6   7   8   9  10

"left" is similar but to the left of the Y axis, ie. into negative X.

    align=>"left"

    26  27  28  29  30  31  32  33  34  35  36        5
            17  18  19  20  21  22  23  24  25        4
                    10  11  12  13  14  15  16        3
                             5   6   7   8   9        2
                                     2   3   4        1
                                             1    <- Y=0

    -10 -9  -8  -7  -6  -5  -4  -3  -2  -1  X=0

The step parameter still controls how much longer each row is than its predecessor.

N Start

The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same rows sequence. For example to start at 0,

    n_start => 0

    16 17 18 19 20 21 22 23 24        4 
        9 10 11 12 13 14 15           3 
           4  5  6  7  8              2 
              1  2  3                 1 
                 0                <- Y=0
    --------------------------
    -4 -3 -2 -1 X=0 1  2  3  4

Step 3 Pentagonals

For step=3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are at the rightmost end of each row. The second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are the vertical at x=-1. Those second numbers are obtained by P(-k), and the two together are the "generalized pentagonal numbers".

Both these sequences are composites from 12 and 15 onwards, respectively, and the immediately preceding P(k)-1, P(k)-2, and S(k)-1, S(k)-2 are too. They factorize simply as

    P(k)   = (3*k-1)*k/2
    P(k)-1 = (3*k+2)*(k-1)/2
    P(k)-2 = (3*k-4)*(k-1)/2
    S(k)   = (3*k+1)*k/2
    S(k)-1 = (3*k-2)*(k+1)/2
    S(k)-2 = (3*k+4)*(k-1)/2

Plotting the primes on a step=3 PyramidRows has the second pentagonal S(k),S(k)-1,S(k)-2 as a 3-wide vertical gap of no primes at X=-1,-2,-3. The the plain pentagonal P(k),P(k-1),P(k)-2 are the endmost three N of each row non-prime. The vertical is much more noticeable in a plot.

       no primes these three columns         no primes these end three
         except the low 2,7,13                     except low 3,5,11
               |  |  |                                /  /  /
     52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
        36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
           23 24 25 26 27 28 29 30 31 32 33 34 35
              13 14 15 16 17 18 19 20 21 22
                  6  7  8  9 10 11 12
                     2  3  4  5
                        1
     -6 -5 -4 -3 -2 -1 X=0 1  2  3  4  5  6  7  8  9 10 11 ...

With align="left" the end values can be put into columns,

                                no primes these end three
    align => "left"                  except low 3,5,11
                                            |  |  |
    36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51        5
             23 24 25 26 27 28 29 30 31 32 33 34 35        4
                      13 14 15 16 17 18 19 20 21 22        3
                                6  7  8  9 10 11 12        2
                                         2  3  4  5        1
                                                  1    <- Y=0
              ... -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0

In general a constant offset S(k)-c is a column and from P(k)-c is a diagonal sloping up dX=2,dY=1 right. The simple factorizations above using the roots of the quadratic P(k)-c or S(k)-c is possible whenever 24*c+1 is a perfect square. This means the further columns S(k)-5, S(k)-7, S(k)-12, etc also have no primes.

The columns S(k), S(k)-1, S(k)-2 are prominent because they're adjacent. There's no other adjacent columns of this type because the squares after 49 are too far apart for 24*c+1 to be a square for successive c. Of course there could be other reasons for other columns or diagonals to have few or many primes.

FUNCTIONS ^

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

$path = Math::PlanePath::PyramidRows->new ()
$path = Math::PlanePath::PyramidRows->new (step => $integer, align => $str, n_start => $n)

Create and return a new path object. The default step is 2. align is a string, one of

    "centre"    the default
    "right"     points aligned right of the Y axis
    "left"      points aligned left of the Y axis

Points are always numbered from left to right in the rows, the alignment changes where each row begins (or ends).

($x,$y) = $path->n_to_xy ($n)

Return the X,Y coordinates of point number $n on the path.

For $n <= 0 the return is an empty list since the path starts at N=1.

$n = $path->xy_to_n ($x,$y)

Return the point number for coordinates $x,$y. $x and $y are each rounded to the nearest integer, which has the effect of treating each point in the pyramid as a square of side 1. If $x,$y is outside the pyramid the return is undef.

($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)

The returned range is exact, meaning $n_lo and $n_hi are the smallest and biggest in the rectangle.

Descriptive Methods

$x = $path->sumxy_minimum()
$x = $path->sumxy_maximum()

Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return undef.

The path is right and above the X=-Y diagonal, thus giving a minimum sum, in the following cases.

    align      condition for sumxy_minimum=0
    ------     -----------------------------
    centre              step <= 3
    right               always
    left                step <= 1
$x = $path->diffxy_minimum()
$x = $path->diffxy_maximum()

Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return undef.

The path is left and above the X=Y leading diagonal, thus giving a minimum X-Y difference, in the following cases.

    align      condition for diffxy_minimum=0
    ------     -----------------------------
    centre              step <= 2
    right               step <= 1
    left                always

OEIS ^

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

http://oeis.org/A023531 (etc)

    step=1
      A002262    X coordinate, runs 0 to k
      A003056    Y coordinate, k repeated k+1 times
      A051162    X+Y sum
      A025581    Y-X diff, runs k to 0
      A079904    X*Y product
      A069011    X^2+Y^2, n_to_rsquared()
      A080099    X bitwise-AND Y
      A080098    X bitwise-OR  Y
      A051933    X bitwise-XOR Y
      A050873    GCD(X+1,Y+1) greatest common divisor by rows
      A051173    LCM(X+1,Y+1) least common multiple by rows

      A023531    dY, being 1 at triangular numbers (but starting n=0)
      A167407    dX-dY, change in X-Y (extra initial 0)
      A129184    turn 1=left, 0=right or straight

      A079824    N total along each opposite diagonal
      A000124    N on Y axis (triangular+1)
      A000217    N on X=Y diagonal, extra initial 0
    step=1, n_start=0
      A109004    GCD(X,Y) greatest common divisor starting (0,0)
      A103451    turn 1=left or right,0=straight, but extra initial 1
      A103452    turn 1=left,0=straight,-1=right, but extra initial 1

    step=2
      A196199    X coordinate, runs -n to +n
      A000196    Y coordinate, n appears 2n+1 times
      A053186    X+Y, being distance to next higher square
      A010052    dY,  being 1 at perfect square row end
      A000290    N on X=Y diagonal, extra initial 0
      A002522    N on X=-Y North-West diagonal (start row), Y^2+1
      A004201    N for which X>=0, ie. right hand half
      A020703    permutation N at -X,Y
    step=2, n_start=0
      A005563    N on X=Y diagonal, Y*(Y+2)
      A000290    N on X=-Y North-West diagonal (start row), Y^2
    step=2, n_start=2
      A059100    N on north-west diagonal (start each row), Y^2+2
      A053615    abs(X), runs k..0..k
    step=2, align=right, n_start=0
      A196199    X-Y, runs -k to +k
      A053615    abs(X-Y), runs k..0..k
    step=2, align=left, n_start=0
      A005563    N on Y axis, Y*(Y+2)
    
    step=3
      A180447    Y coordinate, n appears 3n+1 times
      A104249    N on Y axis, Y*(3Y+1)/2+1
      A143689    N on X=-Y North-West diagonal
    step=3, n_start=0
      A005449    N on Y axis, second pentagonals Y*(3Y+1)/2
      A000326    N on diagonal north-west, pentagonals Y*(3Y-1)/2

    step=4
      A084849    N on Y axis
      A001844    N on X=Y diagonal (North-East)
      A058331    N on X=-Y North-West diagonal
      A221217    permutation N at -X,Y
    step=4, n_start=0
      A014105    N on Y axis, the second hexagonal numbers
      A046092    N on X=Y diagonal, 4*triangular numbers
    step=4, align=right, n_start=0
      A060511    X coordinate, amount n exceeds hexagonal number
      A000384    N on Y axis, the hexagonal numbers
      A001105    N on X=Y diagonal, 2*squares

    step=5
      A116668    N on Y axis

    step=6
      A056108    N on Y axis
      A056109    N on X=Y diagonal (North-East)
      A056107    N on X=-Y North-West diagonal

    step=8
      A053755    N on X=-Y North-West diagonal

    step=9
      A006137    N on Y axis
      A038764    N on X=Y diagonal (North-East)

SEE ALSO ^

Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::Corner, Math::PlanePath::SacksSpiral, Math::PlanePath::MultipleRings

Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Rows

HOME PAGE ^

http://user42.tuxfamily.org/math-planepath/index.html

LICENSE ^

Copyright 2010, 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/>.

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