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
Examples
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;
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.
Crossrefs
See A004481 for the full table.
Programs
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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 *)
Extensions
More terms from Georg Fischer, Feb 22 2020