Provided by: libmath-planepath-perl_125-1_all 
NAME
Math::PlanePath::PyramidSides -- points along the sides of pyramid
SYNOPSIS
use Math::PlanePath::PyramidSides;
my $path = Math::PlanePath::PyramidSides->new;
my ($x, $y) = $path->n_to_xy (123);
DESCRIPTION
This path puts points in layers along the sides of a pyramid growing upwards.
21 4
20 13 22 3
19 12 7 14 23 2
18 11 6 3 8 15 24 1
17 10 5 2 1 4 9 16 25 <- Y=0
------------------------------------
^
... -4 -3 -2 -1 X=0 1 2 3 4 ...
N=1,4,9,16,etc along the positive X axis is the perfect squares. N=2,6,12,20,etc in the
X=-1 vertical is the pronic numbers k*(k+1) half way between those successive squares.
The pattern is the same as the "Corner" path but turned and spread so the single quadrant
in the "Corner" becomes a half-plane here.
The pattern is similar to "PyramidRows" (with its default step=2), just with the columns
dropped down vertically to start at the X axis. Any pattern occurring within a column is
unchanged, but what was a row becomes a diagonal and vice versa.
Lucky Numbers of Euler
An interesting sequence for this path is Euler's k^2+k+41. The low values are spread
around a bit, but from N=1763 (k=41) they're the vertical at X=40. There's quite a few
primes in this quadratic and when plotting primes that vertical stands out a little denser
than its surrounds (at least for up to the first 2500 or so values). The line shows in
other step==2 paths too, but not as clearly. In the "PyramidRows" for instance the
beginning is up at Y=40, and in the "Corner" path it's a diagonal.
N Start
The default is to number points starting N=1 as shown above. An optional "n_start" can
give a different start, in the same pyramid pattern. For example to start at 0,
n_start => 0
20 4
19 12 21 3
18 11 6 13 22 2
17 10 5 2 7 14 23 1
16 9 4 1 0 3 8 15 24 <- Y=0
--------------------------
-4 -3 -2 -1 X=0 1 2 3 4
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
"$path = Math::PlanePath::PyramidSides->new ()"
"$path = Math::PlanePath::PyramidSides->new (n_start => $n)"
Create and return a new path object.
"($x,$y) = $path->n_to_xy ($n)"
Return the X,Y coordinates of point number $n on the path.
For "$n < 0.5" the return is an empty list, it being considered there are no negative
points in the pyramid.
"$n = $path->xy_to_n ($x,$y)"
Return the point number for coordinates "$x,$y". $x and $y are each rounded to the
nearest integer which has the effect of treating points in the pyramid as a squares of
side 1, so the half-plane y>=-0.5 is entirely covered.
"($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.
FORMULAS
Rectangle to N Range
For "rect_to_n_range()", in each column N increases so the biggest N is in the topmost row
and and smallest N in the bottom row.
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.
The smallest N conversely is at the X of smallest absolute value. If the X range crosses
0, ie. $x1 and $x2 have different signs, then X=0 is the smallest.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
<http://oeis.org/A196199> (etc)
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), being floor(sqrt(N)),
k repeated 2k+1 times starting 0
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner,
Math::PlanePath::DiamondSpiral, Math::PlanePath::SacksSpiral, Math::PlanePath::MPeaks
HOME PAGE
<http://user42.tuxfamily.org/math-planepath/index.html>
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017 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/>.