A047708 - OEIS

%I #34 Sep 03 2023 15:24:20

%S 0,2,1,6,7,8,3,5,4,16,14,15,10,9,11,20,13,21,12,25,17,18,19,30,31,38,

%T 35,36,22,23,43,45,48,49,24,26,27,28,29,33,60,32,61,57,66,37,63,34,64,

%U 67,40,39,41,42,82,44,74,79,47,46,87,86,50,95,96,52,101,51,102,53,54

%N Diagonal of Sprague-Grundy function for Wyt Queens (Wythoff's game).

%C 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_.

%C Inverse of sequence A048850 considered as a permutation of the nonnegative integers. - _Howard A. Landman_, Sep 25 2001

%C Comments from _Howard A. Landman_, Nov 24 2007: (Start)

%C 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.

%C 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).

%C 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)

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

%D 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.

%D 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.

%D W. A. Wythoff, "A Modification of the Game of Nim". Nieuw Arch. Wiskunde 8, 199-202, 1907/1909.

%H Rémy Sigrist, <a href="/A047708/b047708.txt">Table of n, a(n) for n = 0..4999</a>

%H H. S. M. Coxeter, <a href="/A000201/a000201.pdf">The Golden Section, Phyllotaxis and Wythoff's Game</a>, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]

%H 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, pp. 249-270 (1999).

%H Rémy Sigrist, <a href="/A047708/a047708.gp.txt">PARI program for A047708</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t mex[list_] :=  mex[list] = Min[Complement[Range[0, Length[list]], list]];

%t move[Wnim, {a_, b_}] :=  move[Wnim, {a, b}] =

%t    Union[Table[{i, b}, {i, 0, a - 1}], Table[{a, i}, {i, 0, b - 1}],

%t     Table[{a - i, b - i}, {i, 1, Min[a, b]}]];

%t SpragueGrundy[game_, list_] :=  SpragueGrundy[game, list] =

%t    mex[SpragueGrundy[game, #] & /@ move[game, list]];

%t Table[SpragueGrundy[Wnim, {i, i}], {i, 0, 64}] (* _Birkas Gyorgy_, Apr 19 2011 *)

%o (PARI) See Links section.

%Y Main diagonal of square array in A004481. Sequences A000201 and A001950 give the m and n coordinates of the zeros of Wythoff (i.e., the P-positions of the game, where the previous player has won).

%Y Cf. A048850.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Howard A. Landman_