cp's OEIS Frontend

A236305 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.

%I A236305 #30 Nov 03 2022 08:20:09
%S A236305 1,4,7,16,19,28,43,64,67,76,91,112,139,172,211,256,259,268,283,304,
%T A236305 331,364,403,448,499,556,619,688,763,844,931,1024,1027,1036,1051,1072,
%U A236305 1099,1132,1171,1216,1267,1324,1387,1456,1531,1612,1699
%N A236305 The number of P-positions in the game of Nim with up to 3 piles, allowing for piles of zero, such that the number of objects in each pile does not exceed n.
%C A236305 P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero.
%C A236305 (0,1,1) is considered different from (1,0,1) and (1,1,0).
%C A236305 a(2^n-1) = 2^(2*n).
%C A236305 Partial sums of A241717.
%C A236305 This sequence seems to be A256534(n+1)/4. - _Thomas Baruchel_, May 15 2018
%H A236305 Michael De Vlieger, <a href="/A236305/b236305.txt">Table of n, a(n) for n = 0..1024</a>
%H A236305 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, p. 37.
%H A236305 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 7 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A236305 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^(2*b) + 3*c^2.
%F A236305 a(n) = 4^floor(log(n)/log(2)) + 3*(n mod 2^floor(log(n)/log(2)))^2 (conjectured). - _Thomas Baruchel_, May 15 2018
%e A236305 If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=4.
%t A236305 Table[Length[Select[Flatten[Table[{n, k, BitXor[n, k]}, {n, 0, a}, {k, 0, a}], 1], #[[3]] <= a &]], {a, 0, 100}]
%Y A236305 Cf. A241522 (4 piles), A241523 (5 piles).
%Y A236305 Cf. A241717 (first differences).
%K A236305 nonn
%O A236305 0,2
%A A236305 _Tanya Khovanova_ and _Joshua Xiong_, Apr 21 2014