++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::PlanePath::DiagonalsAlternating -- points in diagonal stripes of alternating directions

# SYNOPSIS

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

# DESCRIPTION

This path follows successive diagonals going from the Y axis down to the X axis and then back again,

``````      7  |  29
6  |  28  30
5  |  16  27  31
4  |  15  17  26  ...
3  |   7  14  18  25
2  |   6   8  13  19  24
1  |   2   5   9  12  20  23
Y=0  |   1   3   4  10  11  21  22
+----------------------------
X=0   1   2   3   4   5   6``````

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).

## 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,

``````    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  ``````

# FUNCTIONS

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

`\$path = Math::PlanePath::DiagonalsAlternating->new ()`
`\$path = Math::PlanePath::DiagonalsAlternating->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 < 1` the return is an empty list, it being considered the path begins at 1.

# FORMULAS

## 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.

``````    |               N max
|     ----------+
|    |  ^       |
|    |  |       |
|    |   ---->  |
|    +----------
|   N min
+-------------------``````

# OEIS

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

``````    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
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``````

The coordinates such as A003056 X+Y are the same here as in the Diagonals path. `DiagonalsAlternating` transposes X,Y -> Y,X in every second diagonal but forms such as X+Y are unchanged by swapping to Y+X.

http://user42.tuxfamily.org/math-planepath/index.html