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

0,4

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.

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

Table of n, a(n) for n=0..90.

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

Example

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]
...

Prog

(PARI) See Links section.

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]

Sequence in context: A292373 A305383 A004481 * A004489 A305384 A112599

Adjacent sequences: A307293 A307294 A307295 * A307297 A307298 A307299

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 12 2019

Status

approved