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.IX Title "Math::PlanePath::StaircaseAlternating 3pm"
.TH Math::PlanePath::StaircaseAlternating 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
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.SH "NAME"
Math::PlanePath::StaircaseAlternating \-\- stair\-step diagonals up and down
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::StaircaseAlternating;
\& my $path = Math::PlanePath::StaircaseAlternating\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path makes a staircase pattern up from Y axis down to the X and then
back up again.
.PP
.Vb 10
\&    10       46
\&              |
\&     9       47\-\-48
\&                  |
\&     8       45  49\-\-50
\&              |       |
\&     7       44\-\-43  51\-\-52
\&                  |       |
\&     6       16  42\-\-41  53\-\-54
\&              |       |       |
\&     5       17\-\-18  40\-\-39  55\-\-...
\&                  |       |
\&     4       15  19\-\-20  38\-\-37
\&              |       |       |
\&     3       14\-\-13  21\-\-22  36\-\-35
\&                  |       |       |
\&     2        2  12\-\-11  23\-\-24  34\-\-33
\&              |       |       |       |
\&     1        3\-\- 4  10\-\- 9  25\-\-26  32\-\-31
\&                  |       |       |       |
\&    Y=0 \->    1   5\-\- 6   8\-\- 7  27\-\-28  30\-\-29
\&
\&              ^
\&             X=0  1   2   3   4   5   6   7   8
.Ve
.SS "Square Ends"
.IX Subsection "Square Ends"
Option \f(CW\*(C`end_type => "square"\*(C'\fR changes the path as follows, omitting one
point at each end so as to square up the joins.
.PP
.Vb 10
\&     9       42\-\-43
\&              |   |
\&     8       41  44\-\-45
\&              |       |
\&     7       40\-\-39  46\-\-47
\&                  |       |
\&     6        .  38\-\-37  48\-\-49
\&                      |       |
\&     5       14\-\-15  36\-\-35  50\-\-...
\&              |   |       |
\&     4       13  16\-\-17  34\-\-33
\&              |       |       |
\&     3       12\-\-11  18\-\-19  32\-\-31
\&                  |       |       |
\&     2        .  10\-\- 9  20\-\-21  30\-\-29
\&                      |       |       |
\&     1        2\-\- 3   8\-\- 7  22\-\-23  28\-\-27
\&              |   |       |       |       |
\&    Y=0 \->    1   4\-\- 5\-\- 6   .  24\-\-25\-\-26
\&
\&              ^
\&             X=0  1   2   3   4   5   6   7   8
.Ve
.PP
The effect is to shorten each diagonal by a constant 1 each time.  The
lengths of each diagonal still grow by +4 each time (or by +16 up and back).
.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 pattern.  For example to
start at 0,
.PP
.Vb 1
\&    n_start => 0                  n_start => 0, end_type=>"square"
\&
\&    46 47                            41 42
\&    44 48 49                         40 43 44
\&    43 42 50 51                      39 38 45 46
\&    15 41 40 52 53                      37 36 47 48
\&    16 17 39 38 ...                  13 14 35 34 ...
\&    14 18 19 37 36                   12 15 16 33 32
\&    13 12 20 21 35 34                11 10 17 18 31 30
\&     1 11 10 22 23 33 32                 9  8 19 20 29 28
\&     2  3  9  8 24 25 31 30           1  2  7  6 21 22 27 26
\&     0  4  5  7  6 26 27 29 28        0  3  4  5    23 24 25
.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::StaircaseAlternating\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::StaircaseAlternating\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::StaircaseAlternating->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::StaircaseAlternating\->new (end_type => $str, n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::StaircaseAlternating\->new (end_type => $str, n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::StaircaseAlternating->new (end_type => $str, n_start => $n)"
.PD
Create and return a new path object.  The \f(CW\*(C`end_type\*(C'\fR choices are
.Sp
.Vb 2
\&    "jump"        (the default)
\&    "square"
.Ve
.ie n .IP """($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.
.SH "OEIS"
.IX Header "OEIS"
Entries in Sloane's Online Encyclopedia of Integer Sequences related to
this path include
.Sp
.RS 4
<http://oeis.org/A084849> (etc)
.RE
.PP
.Vb 6
\&    end_type=jump, n_start=1  (the defaults)
\&      A084849    N on diagonal X=Y
\&    end_type=jump, n_start=0
\&      A014105    N on diagonal X=Y, second hexagonal numbers
\&    end_type=jump, n_start=2
\&      A096376    N on diagonal X=Y
\&
\&    end_type=square, n_start=1
\&      A058331    N on diagonal X=Y, 2*squares+1
\&    end_type=square, n_start=0
\&      A001105    N on diagonal X=Y, 2*squares
.Ve
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::Staircase,
Math::PlanePath::DiagonalsAlternating
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
.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 <http://www.gnu.org/licenses/>.