.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40)
.\"
.\" Standard preamble:
.\" ========================================================================
.de Sp \" Vertical space (when we can't use .PP)
.if t .sp .5v
.if n .sp
..
.de Vb \" Begin verbatim text
.ft CW
.nf
.ne \\$1
..
.de Ve \" End verbatim text
.ft R
.fi
..
.\" Set up some character translations and predefined strings.  \*(-- will
.\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left
.\" double quote, and \*(R" will give a right double quote.  \*(C+ will
.\" give a nicer C++.  Capital omega is used to do unbreakable dashes and
.\" therefore won't be available.  \*(C` and \*(C' expand to `' in nroff,
.\" nothing in troff, for use with C<>.
.tr \(*W-
.ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p'
.ie n \{\
.    ds -- \(*W-
.    ds PI pi
.    if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch
.    if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\"  diablo 12 pitch
.    ds L" ""
.    ds R" ""
.    ds C` ""
.    ds C' ""
'br\}
.el\{\
.    ds -- \|\(em\|
.    ds PI \(*p
.    ds L" ``
.    ds R" ''
.    ds C`
.    ds C'
'br\}
.\"
.\" Escape single quotes in literal strings from groff's Unicode transform.
.ie \n(.g .ds Aq \(aq
.el       .ds Aq '
.\"
.\" If the F register is >0, we'll generate index entries on stderr for
.\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index
.\" entries marked with X<> in POD.  Of course, you'll have to process the
.\" output yourself in some meaningful fashion.
.\"
.\" Avoid warning from groff about undefined register 'F'.
.de IX
..
.nr rF 0
.if \n(.g .if rF .nr rF 1
.if (\n(rF:(\n(.g==0)) \{\
.    if \nF \{\
.        de IX
.        tm Index:\\$1\t\\n%\t"\\$2"
..
.        if !\nF==2 \{\
.            nr % 0
.            nr F 2
.        \}
.    \}
.\}
.rr rF
.\" ========================================================================
.\"
.IX Title "Math::PlanePath::DiagonalsAlternating 3pm"
.TH Math::PlanePath::DiagonalsAlternating 3pm "2021-01-23" "perl v5.32.0" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
.if n .ad l
.nh
.SH "NAME"
Math::PlanePath::DiagonalsAlternating \-\- points in diagonal stripes of alternating directions
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 3
\& use Math::PlanePath::DiagonalsAlternating;
\& my $path = Math::PlanePath::DiagonalsAlternating\->new;
\& my ($x, $y) = $path\->n_to_xy (123);
.Ve
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
This path follows successive diagonals going from the Y axis down to the X
axis and then back up again,
.PP
.Vb 10
\&      5  |  16
\&         |   |\e
\&      4  |  15  17
\&         |    \e   \e
\&      3  |   7  14  18
\&         |   |\e   \e   \e
\&      2  |   6   8  13  19  ...
\&         |    \e   \e   \e   \e   \e
\&      1  |   2   5   9  12  20  23
\&         |   |\e   \e   \e   \e   \e   \e
\&    Y=0  |   1   3\-\- 4  10\-\-11  21\-\-22
\&         +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&           X=0   1   2   3   4   5   6
.Ve
.PP
The triangular numbers 1,3,6,10,etc k*(k+1)/2 are the
start of each run up or down alternately on the X axis and Y axis.
N=1,6,15,28,etc on the Y axis (Y even) are the hexagonal
numbers j*(2j\-1).  N=3,10,21,36,etc on the X axis (X odd) are the hexagonal
numbers of the second kind j*(2j+1).
.IX Xref "Triangular numbers Hexagonal numbers"
.SS "N Start"
.IX Subsection "N Start"
The default is to number points starting N=1 as shown above.  An optional
\&\f(CW\*(C`n_start\*(C'\fR can give a different start, in the same pattern.  For example to
start at 0,
.PP
.Vb 1
\&    n_start => 0
\&
\&      4  |  14
\&      3  |   6 13
\&      2  |   5  7 12
\&      1  |   1  4  8 11
\&    Y=0  |   0  2  3  9 10
\&         +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
\&           X=0  1  2  3  4
.Ve
.SH "FUNCTIONS"
.IX Header "FUNCTIONS"
See \*(L"\s-1FUNCTIONS\*(R"\s0 in Math::PlanePath for behaviour common to all path classes.
.ie n .IP """$path = Math::PlanePath::DiagonalsAlternating\->new ()""" 4
.el .IP "\f(CW$path = Math::PlanePath::DiagonalsAlternating\->new ()\fR" 4
.IX Item "$path = Math::PlanePath::DiagonalsAlternating->new ()"
.PD 0
.ie n .IP """$path = Math::PlanePath::DiagonalsAlternating\->new (n_start => $n)""" 4
.el .IP "\f(CW$path = Math::PlanePath::DiagonalsAlternating\->new (n_start => $n)\fR" 4
.IX Item "$path = Math::PlanePath::DiagonalsAlternating->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 < 1\*(C'\fR the return is an empty list, it being considered the path
begins at 1.
.SH "FORMULAS"
.IX Header "FORMULAS"
.SS "Rectangle to N Range"
.IX Subsection "Rectangle to N Range"
Within each row increasing X is increasing N, and in each column increasing
Y is increasing N.  So in a rectangle the lower left corner is the minimum N
and the upper right is the maximum N.
.PP
.Vb 8
\&    |               N max
\&    |     \-\-\-\-\-\-\-\-\-\-+
\&    |    |  ^       |
\&    |    |  |       |
\&    |    |   \-\-\-\->  |
\&    |    +\-\-\-\-\-\-\-\-\-\-
\&    |   N min
\&    +\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-
.Ve
.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/A131179> (etc)
.RE
.PP
.Vb 5
\&    n_start=1
\&      A131179    N on X axis (extra initial 0)
\&      A128918    N on Y axis (extra initial 1)
\&      A001844    N on X=Y diagonal
\&      A038722    permutation N at transpose Y,X
\&
\&    n_start=0
\&      A319572    X coordinate
\&      A319573    Y coordinate
\&      A319571    X,Y coordinates together
\&      A003056    X+Y
\&      A004247    X*Y
\&      A049581    abs(X\-Y)
\&      A048147    X^2+Y^2
\&      A004198    X bit\-and Y
\&      A003986    X bit\-or Y
\&      A003987    X bit\-xor Y
\&      A004197    min(X,Y)
\&      A003984    max(X,Y)
\&      A101080    HammingDist(X,Y)
\&      A023531    dSum = dX+dY, being 1 at N=triangular+1 (and 0)
\&      A046092    N on X=Y diagonal
\&      A061579    permutation N at transpose Y,X
\&
\&      A056011    permutation N at points by Diagonals,direction=up order
\&      A056023    permutation N at points by Diagonals,direction=down
\&         runs alternately up and down, both are self\-inverse
.Ve
.PP
The coordinates such as A003056 X+Y are the same here as in the Diagonals
path.  \f(CW\*(C`DiagonalsAlternating\*(C'\fR transposes X,Y \-> Y,X in every second
diagonal but forms such as X+Y are unchanged by swapping to Y+X.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
Math::PlanePath,
Math::PlanePath::Diagonals,
Math::PlanePath::DiagonalsOctant
.SH "HOME PAGE"
.IX Header "HOME PAGE"
<http://user42.tuxfamily.org/math\-planepath/index.html>
.SH "LICENSE"
.IX Header "LICENSE"
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 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 <http://www.gnu.org/licenses/>.