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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. 5
1, 8, 21, 64, 89, 168, 301, 512, 561, 712, 965, 1344, 1801, 2408, 3165, 4096, 4193, 4488, 4981, 5696, 6585, 7720, 9101, 10752, 12433, 14408, 16677, 19264, 22121, 25320, 28861, 32768, 32961, 33544, 34517, 35904, 37657, 39848, 42477, 45568, 48881, 52680, 56965, 61760, 67017 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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).

a(2^n-1) = 2^(3*n).

Partial sums of A241718.

LINKS

Table of n, a(n) for n=0..44.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 8 and J. Int. Seq. 17 (2014) # 14.7.8.

FORMULA

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).

EXAMPLE

If the largest number is 1, then there should be an even number of piles of size 1. Thus, a(1)=8.

MATHEMATICA

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}]

CROSSREFS

Cf. A236305 (3 piles), A241523 (5 piles).

Cf. A241718 (first differences).

Sequence in context: A301538 A192299 A080144 * A096018 A297647 A267144

Adjacent sequences:  A241519 A241520 A241521 * A241523 A241524 A241525

KEYWORD

nonn

AUTHOR

Tanya Khovanova and Joshua Xiong, Apr 24 2014

STATUS

approved

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Last modified October 15 17:58 EDT 2021. Contains 348033 sequences. (Running on oeis4.)