A307296 Array read by antidiagonals: Sprague-Grundy values for the game NimHof with 4 rules [1,0], [3,2], [1,1], [0,1].

0, 1, 1, 2, 2, 2, 3, 0, 0, 3, 4, 4, 1, 4, 4, 5, 5, 5, 5, 5, 5, 6, 3, 3, 6, 3, 3, 6, 7, 7, 4, 7, 7, 4, 7, 7, 8, 8, 8, 1, 8, 0, 8, 8, 8, 9, 6, 6, 0, 2, 2, 1, 6, 6, 9, 10, 10, 7, 2, 9, 7, 9, 2, 7, 10, 10, 11, 11, 11, 9, 10, 10, 10, 0, 9, 11, 11, 11, 12, 9, 9, 12, 0, 11, 3, 11, 1, 12, 9, 9, 12

Offset: 0

Author

N. J. A. Sloane, Apr 12 2019

Keywords

• nonn
• tabf

Comments

The game NimHof with a list of rules R means that for each rule [a,b] you can move from cell [x,y] to any cell [x-i*a,y-i*b] as long as neither coordinate is negative. See the Friedman et al. article for further details.

Examples

			The initial antidiagonals are:
  [0],
  [1, 1],
  [2, 2, 2],
  [3, 0, 0, 3],
  [4, 4, 1, 4, 4],
  [5, 5, 5, 5, 5, 5],
  [6, 3, 3, 6, 3, 3, 6],
  [7, 7, 4, 7, 7, 4, 7, 7],
  [8, 8, 8, 1, 8, 0, 8, 8, 8],
  [9, 6, 6, 0, 2, 2, 1, 6, 6, 9],
  [10, 10, 7, 2, 9, 7, 9, 2, 7, 10, 10],
  [11, 11, 11, 9, 10, 10, 10, 0, 9, 11, 11, 11],
  [12, 9, 9, 12, 0, 11, 3, 11, 1, 12, 9, 9, 12],
The triangle begins:
  [1, 2, 0, 4, 5, 3, 7, 8, 6, 10, 11, 9]
  [2, 0, 1, 5, 3, 4, 8, 6, 7, 11, 9]
  [3, 4, 5, 6, 7, 0, 1, 2, 9, 12]
  [4, 5, 3, 7, 8, 2, 9, 0, 1]
  [5, 3, 4, 1, 2, 7, 10, 11]
  [6, 7, 8, 0, 9, 10, 3]
  [7, 8, 6, 2, 10, 11]
  [8, 6, 7, 9, 0]
  [9, 10, 11, 12]
  [10, 11, 9]
  [11, 9]
  [12]
  ...
		

References

• Eric Friedman, Scott M. Garrabrant, Ilona K. Phipps-Morgan, A. S. Landsberg and Urban Larsson, Geometric analysis of a generalized Wythoff game, in Games of no Chance 5, MSRI publ. Cambridge University Press, date?

Links

• Rémy Sigrist, Colored representation of T(x,y) for x = 0..999 and y = 0..999 (where the hue is function of T(x,y) and black pixels correspond to zeros)
• Rémy Sigrist, PARI program for A307296
• N. J. A. Sloane, Maple program for NimHof sequences

Crossrefs

List of NimHof sequences:

A-number Rules R

A003987 [1,0], [0,1]

A004481 [1,0], [1,1], [0,1]

A003987 [1,0], [2,1], [0,1]

A307300 [1,0], [2,2], [0,1]

A307301 [1,0], [3,1], [0,1]

A003987 [1,0], [3,2], [0,1]

A307302 [1,0], [3,3], [0,1]

A307299 [1,0], [1,1], [1,2], [0,1]

A307296 [1,0], [1,1], [3,2], [0,1]

A307297 [1,0], [2,1], [3,3], [0,1]

A307298 [1,0], [1,1], [1,2], [2,3], [0,1]

Programs

PARI

\\ See Links section.