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A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals. 11
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 (list; table; graph; refs; listen; history; text; internal format)
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.

Blass, Uri, and Aviezri S. Fraenkel. "The Sprague-Grundy function for Wythoff's game." Theoretical Computer Science75.3 (1990): 311-333. See Table 2.

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

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.

EXAMPLE

Table begins

0 1 2 3 4 5 6 7 ...

1 2 0 4 5 3 7 ...

2 0 1 5 3 4 ...

3 4 5 6 2 0 ...

4 5 3 2 ...

5 3 4 ...

6 7 ...

7 ...

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 *)

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 * A004489 A305384 A112599

Adjacent sequences:  A004478 A004479 A004480 * A004482 A004483 A004484

KEYWORD

nonn,tabl,easy,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 16 00:25 EDT 2018. Contains 313782 sequences. (Running on oeis4.)