OFFSET
0,2
COMMENTS
Since Wythoff(m,n) <= m+n, Wythoff(n,n) <= 2n. It is not known whether there is an efficient (linear in log(m)+log(n)) strategy to compute Wythoff(m,n). Each single row is "easy" in the sense that a+n-Wythoff(a,n) is eventually periodic. - Howard A. Landman.
Inverse of sequence A048850 considered as a permutation of the nonnegative integers. - Howard A. Landman, Sep 25 2001
Comments from Howard A. Landman, Nov 24 2007: (Start)
It is impossible for any integer to appear twice in this sequence because of the way it is constructed. Thus to prove that it is a permutation of the integers, we need only show that every value g appears at least once.
Suppose this were not true; then there must be some g such that for any value of n, G(n,n) is not = g. Since G(n,n) is defined to be the smallest number not found as a G(k,n), G(n,k), or G(k,k) for k < n, this can only happen in one of 2 ways: either there is a number g' smaller than g which is chosen (this can occur at most g times) or g already appears as both G(n,k) and G(k,n) for some k < n (because G(n,k) = G(k,n)) (this can happen at most n/2 times).
Thus we have n <= n/2 + g, or n <= 2g; if g has not appeared within the first 2g terms we have a contradiction. Therefore not only must every integer g appear in the sequence, but it must appear within the first 2g terms (and no sooner than term g/2, since G(n,n) <= 2n). Conversely, this also proves that n/2 <= A(n) = G(n,n) <= 2n. (End)
REFERENCES
E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 76.
Howard A. Landman, "A Simple FSM-Based Proof of the Additive Periodicity of the Sprague-Grundy Function of Wythoff's Game", in R. Nowakowski (ed.), More Games of No Chance.
Howard A. Landman and Tom Ferguson showed that this is a permutation of the integers at the Jul 24-28 2000 MSRI workshop on combinatorial games.
W. A. Wythoff, "A Modification of the Game of Nim". Nieuw Arch. Wiskunde 8, 199-202, 1907/1909.
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..4999
H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]
A. Dress, A. Flammenkamp and N. Pink, Additive periodicity of the Sprague-Grundy function of certain Nim games, Adv. Appl. Math., 22, pp. 249-270 (1999).
Rémy Sigrist, PARI program for A047708
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]];
Table[SpragueGrundy[Wnim, {i, i}], {i, 0, 64}] (* Birkas Gyorgy, Apr 19 2011 *)
PROG
(PARI) See Links section.
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Howard A. Landman
STATUS
approved