cp's OEIS Frontend

A241523 The number of P-positions in the game of Nim with up to 5 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

%I A241523 #19 Feb 28 2018 15:06:01
%S A241523 1,16,61,256,421,976,2101,4096,4741,6736,10261,15616,23221,33616,
%T A241523 47461,65536,68101,75856,88981,107776,132661,164176,202981,249856,
%U A241523 305701,371536,448501,537856,640981,759376,894661,1048576,1058821,1089616
%N A241523 The number of P-positions in the game of Nim with up to 5 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.
%C A241523 P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero. (0,1,1,0,0) is considered different from (1,0,1,0,0).
%C A241523 a(2^n-1) = 2^(4n).
%H A241523 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 9 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A241523 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^(4*b) + 10*2^(2*b)*c^2 + 5*c^4.
%e A241523 If the largest number is not more than 1, then there should be an even number of piles of size 1. We can choose the first four piles to be either 0 or 1, then the last pile is uniquely defined. Thus, a(1)=16.
%t A241523 Table[Length[Select[Flatten[Table[{n, k, j, i, BitXor[n, k, j, i]}, {n, 0, a}, {k, 0, a}, {j, 0, a}, {i, 0, a}], 3], #[[5]] <= a &]], {a, 0, 35}]
%Y A241523 Cf. A236305 (3 piles), A241522 (4 piles).
%Y A241523 Cf. A241731 (first differences).
%K A241523 nonn
%O A241523 0,2
%A A241523 _Tanya Khovanova_ and _Joshua Xiong_, Apr 24 2014