A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

1, 7, 14, 50, 63, 105, 148, 364, 413, 491, 546, 798, 883, 1141, 1400, 2696, 2961, 3255, 3382, 3850, 3983, 4313, 4620, 6132, 6469, 6979, 7322, 8870, 9387, 10941, 12496, 20272, 21833, 23423, 23982, 25746, 26167, 26929, 27524, 30332, 30933

Offset

0,2

Comments

Partial sums of A237711.

Links

Table of n, a(n) for n=0..40.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 16 and J. Int. Seq. 17 (2014) # 14.7.8.

Formula

a(2n+1) = 7a(n) + a(n-1), a(2n+2) = a(n+1) + 7a(n).

Example

There is a position (0,0,0,0) with a total of zero. There are 6 positions with a total of 2 that are permutations of (0,0,1,1). Therefore, a(1)=7.

Mathematica

Table[Length[
  Select[Flatten[
    Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0,
      a}], 2], Total[#] <= a &]], {a, 0, 100, 2}]

Crossrefs

Cf. A237711 (first differences), A130665 (3 piles), A238147 (5 piles), A241522, A241718.

Sequence in context: A374509 A045759 A166637 * A170918 A033650 A135536

Adjacent sequences: A237683 A237684 A237685 * A237687 A237688 A237689

Keyword

nonn

Author

Tanya Khovanova and Joshua Xiong, May 02 2014

Status

approved