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.

1, 16, 61, 256, 421, 976, 2101, 4096, 4741, 6736, 10261, 15616, 23221, 33616, 47461, 65536, 68101, 75856, 88981, 107776, 132661, 164176, 202981, 249856, 305701, 371536, 448501, 537856, 640981, 759376, 894661, 1048576, 1058821, 1089616

Offset

0,2

Comments

P-positions in the game of Nim are tuples of numbers with a Nim-Sum equal to zero. (0,1,1,0,0) is considered different from (1,0,1,0,0).

a(2^n-1) = 2^(4n).

Links

Table of n, a(n) for n=0..33.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 9 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^(4*b) + 10*2^(2*b)*c^2 + 5*c^4.

Example

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.

Mathematica

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}]

Crossrefs

Cf. A236305 (3 piles), A241522 (4 piles).

Cf. A241731 (first differences).

Sequence in context: A043229 A044009 A317431 * A264632 A373667 A007831

Adjacent sequences: A241520 A241521 A241522 * A241524 A241525 A241526

Keyword

nonn

Author

Tanya Khovanova and Joshua Xiong, Apr 24 2014

Status

approved