A307301 Array read by antidiagonals: Sprague-Grundy values for the game NimHof with rules [1,0], [3,1], [0,1].

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

Offset: 0

Author

N. J. A. Sloane, Apr 13 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, 0, 2]
[3, 3, 3, 3]
[4, 2, 0, 2, 4]
[5, 5, 4, 1, 5, 5]
[6, 4, 1, 0, 6, 4, 6]
[7, 7, 6, 6, 1, 7, 7, 7]
[8, 6, 5, 7, 0, 6, 4, 6, 8]
[9, 9, 8, 8, 2, 2, 8, 5, 9, 9]
[10, 8, 7, 4, 3, 0, 3, 9, 10, 8, 10]
[11, 11, 10, 5, 9, 9, 1, 8, 5, 11, 11, 11]
[12, 10, 9, 11, 10, 1, 0, 3, 7, 7, 8, 10, 12]
...
The triangle begins:
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
[1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10]
[2, 3, 0, 1, 6, 7, 4, 5, 10, 11, 8]
[3, 2, 4, 0, 1, 6, 8, 9, 5, 7]
[4, 5, 1, 6, 0, 2, 3, 8, 7]
[5, 4, 6, 7, 2, 0, 1, 3]
[6, 7, 5, 8, 3, 9, 0]
[7, 6, 8, 4, 9, 1]
[8, 9, 7, 5, 10]
[9, 8, 10, 11]
[10, 11, 9]
[11, 10]
[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 A307301
• N. J. A. Sloane, Maple program for NimHof sequences

Crossrefs

Cf. A003987, A307296, A307297.

Programs

PARI

See Links section.