A241731 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 the largest pile is n.

1, 15, 45, 195, 165, 555, 1125, 1995, 645, 1995, 3525, 5355, 7605, 10395, 13845, 18075, 2565, 7755, 13125, 18795, 24885, 31515, 38805, 46875, 55845, 65835, 76965, 89355, 103125, 118395, 135285, 153915, 10245, 30795, 51525, 72555, 94005, 115995

Offset

0,2

Comments

This is the finite difference of A241523.

Links

Table of n, a(n) for n=0..37.

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)= 10*2^(2*b)*(2*c-1) + 20*c^3 - 30*c^2 + 20*c - 5.

Example

If the largest pile is 1, then there are 10 positions that are permutations of (0,0,0,1,1) plus 5 positions that are permutations of (0,1,1,1,1). Therefore, a(1)=15.

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], Max[#] == a &]], {a, 0, 50}]

Crossrefs

Cf. A241523, A241717 (3 piles), A241718 (4 piles).

Sequence in context: A039450 A127069 A346853 * A095129 A219813 A068513

Adjacent sequences: A241728 A241729 A241730 * A241732 A241733 A241734

Keyword

nonn

Author

Tanya Khovanova and Joshua Xiong, Apr 27 2014

Status

approved