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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.
5
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
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
KEYWORD
nonn
AUTHOR
Tanya Khovanova and Joshua Xiong, Apr 27 2014
STATUS
approved