cp's OEIS Frontend

A241522 The number of P-positions in the game of Nim with up to 4 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

%I A241522 #21 Nov 03 2022 08:43:48
%S A241522 1,8,21,64,89,168,301,512,561,712,965,1344,1801,2408,3165,4096,4193,
%T A241522 4488,4981,5696,6585,7720,9101,10752,12433,14408,16677,19264,22121,
%U A241522 25320,28861,32768,32961,33544,34517,35904,37657,39848,42477,45568,48881,52680,56965,61760,67017
%N A241522 The number of P-positions in the game of Nim with up to 4 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.
%C A241522 P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero. (0,1,1,0) is considered different from (1,0,1,0).
%C A241522 Partial sums of A241718.
%H A241522 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, pp. 42-43.
%H A241522 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 8 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A241522 If b = floor(log_2(n)) is the number of digits in the binary representation of n and c = n + 1 - 2^b, then a(n) = 2^(3*b) + 6*c^2*2^b + a(c-1).
%F A241522 a(2^n-1) = 2^(3*n).
%e A241522 If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=8.
%t A241522 Table[Length[Select[Flatten[Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0, a}], 2], #[[4]] <= a &]], {a, 0, 50}]
%Y A241522 Cf. A236305 (3 piles), A241523 (5 piles).
%Y A241522 Cf. A241718 (first differences).
%K A241522 nonn
%O A241522 0,2
%A A241522 _Tanya Khovanova_ and _Joshua Xiong_, Apr 24 2014