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Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::PlanePath::PyramidSides \-\- points along the sides of pyramid .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 3 \& use Math::PlanePath::PyramidSides; \& my $path = Math::PlanePath::PyramidSides\->new; \& my ($x, $y) = $path\->n_to_xy (123); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" This path puts points in layers along the sides of a pyramid growing upwards. .PP .Vb 8 \& 21 4 \& 20 13 22 3 \& 19 12 7 14 23 2 \& 18 11 6 3 8 15 24 1 \& 17 10 5 2 1 4 9 16 25 <\- Y=0 \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& ^ \& ... \-4 \-3 \-2 \-1 X=0 1 2 3 4 ... .Ve .PP N=1,4,9,16,etc along the positive X axis is the perfect squares. N=2,6,12,20,etc in the X=\-1 vertical is the pronic numbers k*(k+1) half way between those successive squares. .IX Xref "Square numbers Pronic numbers" .PP The pattern is the same as the \f(CW\*(C`Corner\*(C'\fR path but turned and spread so the single quadrant in the \f(CW\*(C`Corner\*(C'\fR becomes a half-plane here. .PP The pattern is similar to \f(CW\*(C`PyramidRows\*(C'\fR (with its default step=2), just with the columns dropped down vertically to start at the X axis. Any pattern occurring within a column is unchanged, but what was a row becomes a diagonal and vice versa. .SS "Lucky Numbers of Euler" .IX Subsection "Lucky Numbers of Euler" An interesting sequence for this path is Euler's k^2+k+41. The low values are spread around a bit, but from N=1763 (k=41) they're the vertical at X=40. There's quite a few primes in this quadratic and when plotting primes that vertical stands out a little denser than its surrounds (at least for up to the first 2500 or so values). The line shows in other step==2 paths too, but not as clearly. In the \f(CW\*(C`PyramidRows\*(C'\fR for instance the beginning is up at Y=40, and in the \f(CW\*(C`Corner\*(C'\fR path it's a diagonal. .SS "N Start" .IX Subsection "N Start" The default is to number points starting N=1 as shown above. An optional \&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pyramid pattern. For example to start at 0, .PP .Vb 1 \& n_start => 0 \& \& 20 4 \& 19 12 21 3 \& 18 11 6 13 22 2 \& 17 10 5 2 7 14 23 1 \& 16 9 4 1 0 3 8 15 24 <\- Y=0 \& \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& \-4 \-3 \-2 \-1 X=0 1 2 3 4 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes. .ie n .IP """$path = Math::PlanePath::PyramidSides\->new ()""" 4 .el .IP "\f(CW$path = Math::PlanePath::PyramidSides\->new ()\fR" 4 .IX Item "$path = Math::PlanePath::PyramidSides->new ()" .PD 0 .ie n .IP """$path = Math::PlanePath::PyramidSides\->new (n_start => $n)""" 4 .el .IP "\f(CW$path = Math::PlanePath::PyramidSides\->new (n_start => $n)\fR" 4 .IX Item "$path = Math::PlanePath::PyramidSides->new (n_start => $n)" .PD Create and return a new path object. .ie n .IP """($x,$y) = $path\->n_to_xy ($n)""" 4 .el .IP "\f(CW($x,$y) = $path\->n_to_xy ($n)\fR" 4 .IX Item "($x,$y) = $path->n_to_xy ($n)" Return the X,Y coordinates of point number \f(CW$n\fR on the path. .Sp For \f(CW\*(C`$n < 0.5\*(C'\fR the return is an empty list, it being considered there are no negative points in the pyramid. .ie n .IP """$n = $path\->xy_to_n ($x,$y)""" 4 .el .IP "\f(CW$n = $path\->xy_to_n ($x,$y)\fR" 4 .IX Item "$n = $path->xy_to_n ($x,$y)" Return the point number for coordinates \f(CW\*(C`$x,$y\*(C'\fR. \f(CW$x\fR and \f(CW$y\fR are each rounded to the nearest integer which has the effect of treating points in the pyramid as a squares of side 1, so the half-plane y>=\-0.5 is entirely covered. .ie n .IP """($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)""" 4 .el .IP "\f(CW($n_lo, $n_hi) = $path\->rect_to_n_range ($x1,$y1, $x2,$y2)\fR" 4 .IX Item "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)" The returned range is exact, meaning \f(CW$n_lo\fR and \f(CW$n_hi\fR are the smallest and biggest in the rectangle. .SH "FORMULAS" .IX Header "FORMULAS" .SS "Rectangle to N Range" .IX Subsection "Rectangle to N Range" For \f(CW\*(C`rect_to_n_range()\*(C'\fR, in each column N increases so the biggest N is in the topmost row and and smallest N in the bottom row. .PP In each row N increases along the sequence X=0,\-1,1,\-2,2,\-3,3, etc. So the biggest N is at the X of biggest absolute value and preferring the positive X=k over the negative X=\-k. .PP The smallest N conversely is at the X of smallest absolute value. If the X range crosses 0, ie. \f(CW$x1\fR and \f(CW$x2\fR have different signs, then X=0 is the smallest. .SH "OEIS" .IX Header "OEIS" Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include .Sp .RS 4 (etc) .RE .PP .Vb 6 \& n_start=1 (the default) \& A049240 abs(dY), being 0=horizontal step at N=square \& A002522 N on X negative axis, x^2+1 \& A033951 N on X=Y diagonal, 4d^2+3d+1 \& A004201 N for which X>=0, ie. right hand half \& A020703 permutation N at \-X,Y \& \& n_start=0 \& A196199 X coordinate, runs \-n to +n \& A053615 abs(X), runs n to 0 to n \& A000196 abs(X)+abs(Y), floor(sqrt(N)), \& k repeated 2k+1 times starting 0 .Ve .SH "SEE ALSO" .IX Header "SEE ALSO" Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner, Math::PlanePath::DiamondSpiral, Math::PlanePath::SacksSpiral, Math::PlanePath::MPeaks .SH "HOME PAGE" .IX Header "HOME PAGE" .SH "LICENSE" .IX Header "LICENSE" Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde .PP This file is part of Math-PlanePath. .PP Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the \s-1GNU\s0 General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. .PP Math-PlanePath is distributed in the hope that it will be useful, but \&\s-1WITHOUT ANY WARRANTY\s0; without even the implied warranty of \s-1MERCHANTABILITY\s0 or \s-1FITNESS FOR A PARTICULAR PURPOSE. \s0 See the \s-1GNU\s0 General Public License for more details. .PP You should have received a copy of the \s-1GNU\s0 General Public License along with Math-PlanePath. If not, see .