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A238147 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 doesn't exceed 2n.

%I A238147 #22 Dec 14 2018 16:09:33
%S A238147 1,11,26,126,191,341,516,1516,2081,2731,3206,4706,5631,7381,9256,
%T A238147 19256,24821,30471,33946,40446,44171,48921,52796,67796,76221,85471,
%U A238147 91846,109346,119971,138721,158096,258096,313661,369311
%N A238147 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 doesn't exceed 2n.
%C A238147 Partial sums of A238759.
%H A238147 T. Khovanova and J. Xiong, <a href="http://arxiv.org/abs/1405.5942">Nim Fractals</a>, arXiv:1405.594291 [math.CO] (2014), p. 18 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Khovanova/khova6.html">J. Int. Seq. 17 (2014) # 14.7.8</a>.
%F A238147 a(2n+1) = 11a(n) + 5a(n-1), a(2n+2) = a(n+1) + 15a(n).
%e A238147 There is 1 position (0,0,0,0,0) with a total of zero. There are 10 positions with a total of 2 that are permutations of (0,0,0,1,1). Therefore, a(1)=11.
%t A238147 Table[Length[
%t A238147   Select[Flatten[
%t A238147     Table[{n, k, j, i, BitXor[n, k, j, i]}, {n, 0, a}, {k, 0, a}, {j,
%t A238147       0, a}, {i, 0, a}], 3], #[[5]] <= a &]], {a, 0, 35}]
%t A238147 (* Second program: *)
%t A238147 a[n_] := a[n] = Which[n <= 1, {1, 11}[[n+1]], OddQ[n], 11 a[(n-1)/2] + 5 a[(n-1)/2 - 1], EvenQ[n], a[(n-2)/2 + 1] + 15*a[(n-2)/2]];
%t A238147 Array[a, 34, 0] (* _Jean-François Alcover_, Dec 14 2018 *)
%Y A238147 Cf. A238759 (first differences), A130665 (3 piles), A237686 (4 piles), A241523, A241731.
%K A238147 nonn
%O A238147 0,2
%A A238147 _Tanya Khovanova_ and _Joshua Xiong_, May 02 2014