A241718 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 the largest pile is n.

1, 7, 13, 43, 25, 79, 133, 211, 49, 151, 253, 379, 457, 607, 757, 931, 97, 295, 493, 715, 889, 1135, 1381, 1651, 1681, 1975, 2269, 2587, 2857, 3199, 3541, 3907, 193, 583, 973, 1387, 1753, 2191, 2629, 3091, 3313, 3799, 4285, 4795, 5257, 5791, 6325

Offset

0,2

Comments

This is the first difference of A241522.

Links

Table of n, a(n) for n=0..46.

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) = (12*c-6)*2^b + a(c-1).

Example

If the largest pile is 2, then there are 6 positions that are permutations of (0,0,2,2) plus 6 positions that are permutations of (1,1,2,2) and one position (2,2,2,2). Therefore, a(2)=13.

Mathematica

Table[Length[Select[Flatten[Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0, a}], 2], Max[#] == a &]], {a, 0, 50}]

Crossrefs

Cf. A241522, A241717 (3 piles), A241731 (5 piles).

Sequence in context: A134854 A330671 A097444 * A259184 A259186 A151781

Adjacent sequences: A241715 A241716 A241717 * A241719 A241720 A241721

Keyword

nonn

Author

Tanya Khovanova and Joshua Xiong, Apr 27 2014

Status

approved