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 ...
The horizontal 1,4,9,16,etc at the bottom going right 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 pattern is the same as the Corner path but widened out so that the single quadrant in the Corner becomes a halfplane here.
The pattern is similar to PyramidRows, 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. Low values are spread around a bit, but from N=1763 (k=41) onwards they're the vertical at x=40. There's quite a few primes in this quadratic and on a plot of the primes that vertical stands out a little denser in primes 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 the beginning is up at y=40, and in the Corner path it's a diagonal.
FUNCTIONS
$path = Math::PlanePath::PyramidSides>new ()

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 halfplane y>=0.5 is entirely covered.
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner, Math::PlanePath::SacksSpiral
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
MathPlanePath is Copyright 2010 Kevin Ryde
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.