Math::...idRows(3pm)

📦

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✨ Math::PlanePath::PyramidRows ✨

from User Contributed Perl Documentation — manual name folded in near the top, cute and cozy 🎀

💖 NAME 💖

Math::PlanePath::PyramidRows — points stacked up in a pyramid 👑

That's the whole name section, short and iconic, periodt.

🎀 SYNOPSIS 🎀

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

🦋 DESCRIPTION 🦋

okay so this path arranges points in successively wider rows going upwards, so it forms an upside-down pyramid. the default step is 2, meaning each row is 2 wider than the preceding one — an extra point at the left and the right. it's giving staircase but make it fashion 💅

           17  18  19  20  21  22  23  24  25       4
               10  11  12  13  14  15  16           3     step => 2
                    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. math girlies stay winning 🔥

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.

           11  12  13  14  15         4
            7   8   9  10             3       step => 1
            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 consistent but unlikely to be much use. The Rows path with "width" 1 does this too.

            5         4
            4         3     step => 0
            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 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 🔎

Okay so Sloane's Online Encyclopedia of Integer Sequences has entries related to this path, and here they all are. The whole index entry lives at:

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

Now let's break it down by config, because every step / n_start / align combo pulls up a different set of bangers 💅

🌈 `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 👀

Go read these next bestie, they're all connected:

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, 2015, 2016, 2017, 2018, 2019, 2020 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/>

and that's the whole page bestie, we did it 🫶✨ this one shipped as part of perl v5.32.0, dated 2021-01-23, filed under Math::...idRows(3pm) — literally iconic, no cap 💅💖

📦 where this page comes from

debian bookworm: libmath-planepath-perl 129-1 · GPL-3.0-or-later
debian trixie: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu jammy: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu noble: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu resolute: libmath-planepath-perl 129-1 · GPL-3.0-or-later
ubuntu stonking: libmath-planepath-perl 129-1 · GPL-3.0-or-later
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash on 2026-10-14: a language model's rewrite of the original, under its terms

Math::...idRows(3pm)

✨ Math::PlanePath::PyramidRows ✨

💖 NAME 💖

🎀 SYNOPSIS 🎀

🦋 DESCRIPTION 🦋

🌈 Step Parameter 🌈

🎀 Align Parameter 🎀

🌸 N Start 🌸

💜 Step 3 Pentagonals 💜

🔥 FUNCTIONS 🔥

✨ Descriptive Methods ✨

🔎 OEIS 🔎

🌈 step=1 🌈

🌸 step=1, n_start=0 🌸

🦋 step=2 🦋

🎀 step=2, n_start=0 🎀

✨ step=2, n_start=2 ✨

💜 step=2, align=right, n_start=0 💜

🍓 step=2, align=left, n_start=0 🍓

🔥 step=3 🔥

🌟 step=3, n_start=0 🌟

🧚 step=4 🧚

💖 step=4, n_start=0 💖

🌈 step=4, align=right, n_start=0 🌈

👑 step=5 👑

⭐ step=6 ⭐

🎀 step=8 🎀

🦋 step=9 🦋