cp's OEIS Frontend

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

%I A004481 #56 Apr 18 2025 03:18:33
%S A004481 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,
%T A004481 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,
%U A004481 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
%N A004481 Table of Sprague-Grundy values for Wythoff's game (Wyt Queens) read by antidiagonals.
%C A004481 T(a,b) = T(b,a).
%D A004481 E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
%D A004481 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?
%D A004481 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.
%H A004481 Vincenzo Librandi, <a href="/A004481/b004481.txt">Table of n, a(n) for n = 0..5049</a>
%H A004481 Uri Blass and Aviezri S. Fraenkel, <a href="https://doi.org/10.1016/0304-3975(90)90098-3">The Sprague-Grundy function for Wythoff's game</a>, Theoretical Computer Science 75.3 (1990): 311-333. See Table 2.
%H A004481 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://doi.org/10.37236/8905">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A004481 A. Dress, A. Flammenkamp and N. Pink, <a href="http://dx.doi.org/10.1006/aama.1998.0632">Additive periodicity of the Sprague-Grundy function of certain Nim games</a>, Adv. Appl. Math., 22, p. 249-270 (1999).
%H A004481 Gabriel Nivasch, <a href="https://library.slmath.org/books/Book56/files/43nivasch.pdf">More on the Sprague-Grundy function for Wythoff's game</a>, pages 377-410 in "Games of No Chance 3", MSRI Publications Volume 56, 2009. See Table 1.
%H A004481 Rémy Sigrist, <a href="/A004481/a004481.png">Colored representation of T(x,y) for x = 0..999 and y = 0..999</a> (where the hue is function of T(x,y) and black pixels correspond to zeros)
%H A004481 Rémy Sigrist, <a href="/A004481/a004481.gp.txt">PARI program for A004481</a>
%e A004481 Table begins
%e A004481    0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
%e A004481    1,  2,  0,  4,  5,  3,  7,  8,  6, 10, 11,  9, ...
%e A004481    2,  0,  1,  5,  3,  4,  8,  6,  7, 11,  9, ...
%e A004481    3,  4,  5,  6,  2,  0,  1,  9, 10, 12, ...
%e A004481    4,  5,  3,  2,  7,  6,  9,  0,  1, ...
%e A004481    5,  3,  4,  0,  6,  8, 10,  1, ...
%e A004481    6,  7,  8,  1,  9, 10,  3, ...
%e A004481    7,  8,  6,  9,  0,  1, ...
%e A004481    8,  6,  7, 10,  1, ...
%e A004481    9, 10, 11, 12, ...
%e A004481   10, 11,  9, ...
%e A004481   11,  9, ...
%e A004481   12, ...
%e A004481   ...
%t A004481 mex[list_] := mex[list] = Min[Complement[Range[0, Length[list]], list]];
%t A004481 move[Wnim, {a_, b_}] := move[Wnim, {a, b}] =
%t A004481    Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],
%t A004481     Table[{a - i, b - i}, {i, 1, Min[a, b]}]];
%t A004481 SpragueGrundy[game_, list_] := SpragueGrundy[game, list] =
%t A004481    mex[SpragueGrundy[game, #] & /@ move[game, list]];
%t A004481 t[n_, m_] := SpragueGrundy[Wnim, {n - 1, m - 1}];
%t A004481 Flatten@Table[t[n - m + 1, m], {n, 11}, {m, n}] (* _Birkas Gyorgy_, Apr 19 2011 *)
%o A004481 (PARI) See Links section.
%Y A004481 A004482-A004487 are rows 1 to 6. Cf. A047708 (main diagonal).
%Y A004481 See A317205 for triangle of values on or below main diagonal.
%Y A004481 Similar to but different from A004489.
%Y A004481 T(a, b)=0 iff A018219(a, b)=0 iff A002251(a)=b.
%K A004481 nonn,tabl,easy,nice
%O A004481 0,4
%A A004481 _N. J. A. Sloane_