Pyramid Rows

Taking two diagonals running from k^2+1 to (k+1)^2 is the same as a row of the step=2 "PyramidRows" (see Math::PlanePath::PyramidRows). Each endpoint is the same, but here it's two diagonals instead of one row. For example in the "PyramidRows" the Y=3 row runs from N=10 to N=16 ending at X=3,Y=3. Here that's in two diagonals N=10 to N=12 and then N=13 to N=16, and that N=16 endpoint is the same X=3,Y=3.

Direction

Option "direction => 'up'" reverses the order within each diagonal and counts upward from the centre to the Y axis.

    8 |  25 29 34 39 45 51 58 65 73 
      |    \  \  \  \  \  \  \
    7 |  20 24 28 33 38 44 50 57
      |    \  \  \  \  \  \
    6 |  16 19 23 27 32 37 43
      |    \  \  \  \  \
    5 |  12 15 18 22 26 31
      |    \  \  \  \
    4 |   9 11 14 17 21             direction => "up"
      |    \  \  \
    3 |   6  8 10 13
      |    \  \
    2 |   4  5  7
      |    \
    1 |   2  3
      |
  Y=0 |   1
      +---------------------------
        X=0  1  2  3  4  5  6  7  8

In this arrangement N=1,2,4,6,9,etc on the Y axis are alternately the squares and the pronic numbers. The squares are on even Y and pronic on odd Y.

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 diagonals sequence. For example to start at 0,

    n_start => 0                    n_start=>0
    direction => "down"             direction=>"up"
      6  | 12                        | 15
      5  |  9 13                     | 11 14
      4  |  6 10 14                  |  8 10 13
      3  |  4  7 11 15               |  5  7  9 12
      2  |  2  5  8                  |  3  4  6
      1  |  1  3                     |  1  2
    Y=0  |  0                        |  0
         +--------------             +--------------
          X=0  1  2  3                X=0  1  2  3

N to X,Y

To break N into X,Y it's convenient to take two diagonals at a time, since the length then changes by 1 each pair making a quadratic. Starting at each X=0,Y=odd just after perfect square N allows just a sqrt.

    Nstart = d*d+1

where d numbers diagonal pairs, eg. d=3 for X=0,Y=5 going down. This is easily reversed as

    d = floor sqrt(N-1)

The code reckons the start of the diagonal as 0.5 further back, so that N=9.5 is at X=-.5,Y=5.5. To do that d is formed as

    d = floor sqrt(N-0.5)
      = int( sqrt(int(4*$n)-2)/2 )

Taking /2 out of the sqrt helps with "Math::BigInt" which circa Perl 5.14 doesn't inter-operate with flonums very well.

In any case N-Nstart is an offset into two diagonals, the first of length d many points and the second d+1. For example d=3 starting Y=5 for points N=10,11,12 followed by Y=6 N=13,14,15,16.

The formulas are simplified by calculating a remainder relative to the second diagonal, so it's negative for the first and positive for the second,

    Nrem = N - (d*(d+1)+1)

d*(d+1)+1 is 1 past the pronic numbers when end each first diagonal, as described above. In any case for example d=3 is relative to N=13 making Nrem=-3,-2,-1 or Nrem=0,1,2,3.

To include the preceding 0.5 in the second diagonal simply means reckoning Nrem>=-0.5 as belonging to the second. In that base

    if Nrem >= -0.5
      X = Nrem            # direction="down"
      Y = 2*d - Nrem
    else
      X = Nrem + d
      Y = d - Nrem - 1

For example N=15 Nrem=1 is the first case, X=1, Y=2*3-1=5. Or N=11 Nrem=-2 the second X=-2+3=1, Y=3-(-2)-1=4.

For "up" direction the Nrem and d are the same, but the coordinate directions reverse.

    if Nrem >= -0.5
      X = d - Nrem        # direction="up"
      Y = d + Nrem
    else
      X = -Nrem - 1
      Y = 2d + Nrem

Another way is to reckon Nstart from the X=0,Y=even diagonals, which is then two diagonals of the same length and d formed by a sqrt inverting a pronic Nstart=d*(d+1).

Rectangle to N Range

Within each row increasing X is increasing N, and in each column increasing Y is increasing N. This is so in both "down" and "up" arrangements. On that basis in a rectangle the lower left corner is the minimum N and the upper right is the maximum N.

If the rectangle is partly outside the covered octant then the corners must be shifted to put them in range, ie. trim off any rows or columns entirely outside the rectangle. For the lower left this means,

      |  |    /
      |  |   /
      +--------          if x1 < 0 then x1 = 0
    x1   | /             increase x1 to within octant
         |/
         +
         |  |/
         |  |            if y1 < x1 then y1 = x1
         | /|            increase y1 to bottom-left within octant
         |/ +----y1
         +  x1

And for the top right,

         |    /  x2
         | ------+ y2    if x2 > y2 then x2 = y2
         |  /    |       decrease x2 so top-right within octant
         | /     |         (the end of the y2 row)
         |/
         +