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A238759 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.

%I A238759 #26 Apr 02 2020 03:10:38
%S A238759 1,10,15,100,65,150,175,1000,565,650,475,1500,925,1750,1875,10000,
%T A238759 5565,5650,3475,6500,3725,4750,3875,15000,8425,9250,6375,17500,10625,
%U A238759 18750,19375,100000,55565,55650,33475,56500,31725,34750,23875,65000
%N A238759 The number of P-positions in the game of Nim with up to five piles, allowing for piles of zero, such that the total number of objects in all piles is 2n.
%C A238759 First differences of A238147.
%H A238759 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 17 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A238759 a(2n+1) = 10*a(n), a(2n+2) = a(n+1) + 5*a(n).
%e A238759 The P-positions with the total of 4 are permutations of (0,0,0,2,2) and (0,1,1,1,1). Therefore, a(2)=15.
%t A238759 Table[Length[
%t A238759   Select[Flatten[
%t A238759     Table[{n, k, j, i, BitXor[n, k, j, i]}, {n, 0, a}, {k, 0, a}, {j,
%t A238759       0, a}, {i, 0, a}], 3], Total[#] == a &]], {a, 0, 90, 2}]
%t A238759 (* Second program: *)
%t A238759 (* b = A238147 *) b[n_] := b[n] = Which[n <= 1, {1, 11}[[n+1]], OddQ[n], 11 b[(n-1)/2] + 5 b[(n-1)/2 - 1], EvenQ[n], b[(n-2)/2 + 1] + 15 b[(n-2)/2]];
%t A238759 Join[{1}, Differences[Array[b, 40, 0]]] (* _Jean-François Alcover_, Dec 14 2018 *)
%Y A238759 Cf. A238147 (partial sums), A048883 (3 piles), A237711 (4 piles), A241523, A241731.
%K A238759 nonn
%O A238759 0,2
%A A238759 _Tanya Khovanova_ and _Joshua Xiong_, May 02 2014