NAME

       Math::PlanePath::PentSpiralSkewed -- integer points in a pentagonal shape

SYNOPSIS

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

DESCRIPTION

       This path makes a pentagonal (five-sided) spiral with points skewed so as to fit a square
       grid and fully cover the plane.

                 10 ...             2
                /  \  \
              11  3  9 20           1
             /  /  \  \  \
           12  4  1--2  8 19    <- Y=0
             \  \       |  |
              13  5--6--7 18       -1
                \          |
                 14-15-16-17       -2

            ^  ^  ^  ^  ^  ^
           -2 -1 X=0 1  2  3 ...

       The pattern is similar to the "SquareSpiral" but cuts three corners which makes each cycle
       is faster.  Each cycle is just 5 steps longer than the previous (where it's 8 for a
       "SquareSpiral").

   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 pattern.  For example to start at 0,

                       38             n_start => 0
                    39 21 37  ...
                 40 22  9 20 36 57
              41 23 10  2  8 19 35 56
           42 24 11  3  0  1  7 18 34 55
              43 25 12  4  5  6 17 33 54
                 44 26 13 14 15 16 32 53
                    45 27 28 29 30 31 52
                       46 47 48 49 50 51

FUNCTIONS

       See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

       "$path = Math::PlanePath::PentSpiral->new ()"
       "$path = Math::PlanePath::PentSpiral->new (n_start => $n)"
           Create and return a new path object.

       "$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 path as a square
           of side 1.

OEIS

       Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

           <http://oeis.org/A192136> (etc)

           n_start=1 (the default)
             A192136    N on X axis, (5*n^2 - 3*n + 2)/2
             A140066    N on Y axis
             A116668    N on X negative axis, (5n^2 + n + 2)/2
             A134238    N on Y negative axis
             A158187    N on North-West diagonal, 10*n^2 + 1
             A005891    N on South-East diagonal, centred pentagonals

           n_start=0
             A000566    N on X axis, heptagonal numbers
             A005476    N on Y axis
             A005475    N on X negative axis
             A147875    N on Y negative axis, second heptagonals
             A033583    N on North-West diagonal, 10*n^2
             A028895    N on South-East diagonal, 5*triangular

SEE ALSO

       Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral,
       Math::PlanePath::HexSpiralSkewed

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/>.