A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.

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

Offset

0,4

Comments

T(a,b) = T(b,a).

References

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

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?

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..5049

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.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

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).

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.

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 A004481

Example

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

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}];
Flatten@Table[t[n - m + 1, m], {n, 11}, {m, n}] (* Birkas Gyorgy, Apr 19 2011 *)

Prog

(PARI) See Links section.

Crossrefs

A004482-A004487 are rows 1 to 6. Cf. A047708 (main diagonal).

See A317205 for triangle of values on or below main diagonal.

Similar to but different from A004489.

T(a, b)=0 iff A018219(a, b)=0 iff A002251(a)=b.

Sequence in context: A035307 A292373 A305383 * A307296 A004489 A305384

Adjacent sequences: A004478 A004479 A004480 * A004482 A004483 A004484

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Status

approved