A307298 Array read by antidiagonals: Sprague-Grundy values for the game NimHof with 5 rules [1,0], [1,1], [2,3], [1,2], [0,1].

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

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, 3, 3]
[4, 4, 4, 0, 4]
[5, 5, 5, 5, 5, 5]
[6, 3, 0, 6, 6, 6, 6]
[7, 7, 1, 7, 0, 7, 7, 7]
[8, 8, 8, 8, 1, 1, 1, 4, 8]
[9, 6, 9, 2, 2, 2, 2, 8, 9, 9]
[10, 10, 10, 1, 3, 9, 3, 9, 3, 10, 10]
[11, 11, 11, 4, 10, 4, 4, 10, 10, 11, 11, 11]
[12, 9, 6, 12, 11, 11, 5, 11, 11, 12, 12, 8, 12]
...
The triangle begins:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
[1, 2, 3, 0, 5, 6, 7, 4, 9, 10, 11, 8]
[2, 0, 4, 5, 6, 7, 1, 8, 3, 11, 12]
[3, 4, 5, 6, 0, 1, 2, 9, 10, 12]
[4, 5, 0, 7, 1, 2, 3, 10, 11]
[5, 3, 1, 8, 2, 9, 4, 11]
[6, 7, 8, 2, 3, 4, 5]
[7, 8, 9, 1, 10, 11]
[8, 6, 10, 4, 11]
[9, 10, 11, 12]
[10, 11, 6]
[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..1023 and y = 0..1023 (where the hue is function of T(x,y) and black pixels correspond to zeros)
• Rémy Sigrist, PARI program for A307298
• N. J. A. Sloane, Maple program for NimHof sequences

Crossrefs

Cf. A003987, A307296, A307297.

Programs

PARI

See Links section.