A351630 - OEIS

%I #84 Jun 02 2022 10:30:42

%S 0,12,1270,105161

%N Nim values that occur at infinitely many heap sizes in the combinatorial game Mem0.

%C The impartial combinatorial game Mem0 (aka Short Local Nim) is played with heaps of tokens, as in Nim. On each turn, k tokens may be removed from a heap H, provided that k is not equal to the number of tokens that were removed on the immediately preceding move on H.

%C A heap may be denoted by n_k, where n is the number of tokens remaining and k the number removed on the preceding move. There are many nim values m that occur at just finitely many heap sizes, in the sense that G(n_k) = m for just finitely many choices of n. This sequence gives the exceptional values of m that occur at infinitely many heap sizes.

%C It is unknown whether there are infinitely many such m. It is remarkable that such simple, parameterless rules give rise to an unusual and mysterious integer sequence.

%D R. K. Guy and R. J. Nowakowski, Unsolved Problems in Combinatorial Games, More Games of No Chance, MSRI Publications, Volume 42, 2002, pp. 457-473, problem 22.

%H Urban Larsson, Simon Rubinstein-Salzedo, and Aaron N. Siegel, <a href="https://arxiv.org/abs/1912.10517">Memgames</a>, arXiv:1912.10517 [math.CO], 2019.

%Y Cf. A131469.

%K nonn,hard,more

%O 0,2

%A _Aaron N. Siegel_, Jun 01 2022