--- name: "Math::PlanePath::PyramidRows" section: "3pm" platform: "linux" package: "libmath-planepath-perl" version: "129-1" release: "Debian 13 (Trixie)" platforms: ["linux"] url: "https://new.carta.tech/Math::PlanePath::PyramidRows" --- # Math::PlanePath::PyramidRows(3pm) ## 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 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. 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 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 \ (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 \ ## 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 \.