A317205 Sprague-Grundy values for Wythoff's game.

0, 1, 2, 2, 0, 1, 3, 4, 5, 6, 4, 5, 3, 2, 7, 5, 3, 4, 0, 6, 8, 6, 7, 8, 1, 9, 10, 3, 7, 8, 6, 9, 0, 1, 4, 5, 8, 6, 7, 10, 1, 2, 5, 3, 4, 9, 10, 11, 12, 8, 7, 13, 14, 15, 16, 10, 11, 9, 8, 13, 12, 0, 15, 16, 17, 14, 11, 9, 10, 7, 12, 14, 2, 13, 17, 6, 18, 15

Offset

0,3

References

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.

Links

Georg Fischer, Table of n, a(n) for n = 0..1274 (First 50 rows)

Uri Blass and Aviezri S. Fraenkel, The Sprague-Grundy function for Wythoff's game, Theoretical Computer Science 75.3 (1990): 311-333. See Table 2.

A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, p. 249-270 (1999).

R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.

Gabriel Nivasch, More on the Sprague-Grundy function for Wythoff’s game, pages 377-410 in "Games of No Chance 3, MSRI Publications Volume 56, 2009. See Table 1.

Example

Triangle begins as:
  0;
  1,  2;
  2,  0,  1;
  3,  4,  5,  6;
  4,  5,  3,  2,  7;
  5,  3,  4,  0,  6,  8;
  6,  7,  8,  1,  9, 10,  3;
  7,  8,  6,  9,  0,  1,  4,  5;

Mathematica

mex[list_] := mex[list] = Min[Complement[Range[0, Length[list]], list]];
move[Wnim, {a_, b_}] := move[Wnim, {a, b}] =
   Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
    Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
SpragueGrundy[game_, list_] := SpragueGrundy[game, list] =
   mex[SpragueGrundy[game, #] & /@ move[game, list]];
t[n_, m_] := SpragueGrundy[Wnim, {n - 1, m - 1}]; (* so far copied from A004481 *)
Flatten[Table[t[n, m], {n, 12}, {m, 1, n}]] (* Georg Fischer, Feb 22 2020 *)

Crossrefs

See A004481 for the full table.

Sequence in context: A321884 A136431 A182888 * A296068 A144064 A172236

Adjacent sequences: A317202 A317203 A317204 * A317206 A317207 A317208

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 07 2018

Extensions

More terms from Georg Fischer, Feb 22 2020

Status

approved