A238147 -id:A238147 - cp's OEIS frontend

A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

1, 7, 14, 50, 63, 105, 148, 364, 413, 491, 546, 798, 883, 1141, 1400, 2696, 2961, 3255, 3382, 3850, 3983, 4313, 4620, 6132, 6469, 6979, 7322, 8870, 9387, 10941, 12496, 20272, 21833, 23423, 23982, 25746, 26167, 26929, 27524, 30332, 30933

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A237711.

			There is a position (0,0,0,0) with a total of zero. There are 6 positions with a total of 2 that are permutations of (0,0,1,1). Therefore, a(1)=7.

T. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 16 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A237711 (first differences), A130665 (3 piles), A238147 (5 piles), A241522, A241718.

Table[Length[
  Select[Flatten[
    Table[{n, k, j, BitXor[n, k, j]}, {n, 0, a}, {k, 0, a}, {j, 0,
      a}], 2], Total[#] <= a &]], {a, 0, 100, 2}]

a(2n+1) = 7a(n) + a(n-1), a(2n+2) = a(n+1) + 7a(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.

Original entry on oeis.org

1, 10, 15, 100, 65, 150, 175, 1000, 565, 650, 475, 1500, 925, 1750, 1875, 10000, 5565, 5650, 3475, 6500, 3725, 4750, 3875, 15000, 8425, 9250, 6375, 17500, 10625, 18750, 19375, 100000, 55565, 55650, 33475, 56500, 31725, 34750, 23875, 65000

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

First differences of A238147.

			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. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 17 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A238147 (partial sums), A048883 (3 piles), A237711 (4 piles), A241523, A241731.

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], Total[#] == a &]], {a, 0, 90, 2}]
(* Second program: *)
(* 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]];
Join[{1}, Differences[Array[b, 40, 0]]] (* Jean-François Alcover, Dec 14 2018 *)

a(2n+1) = 10*a(n), a(2n+2) = a(n+1) + 5*a(n).

cp's OEIS Frontend

A237686 The number of P-positions in the game of Nim with up to four piles, allowing for piles of zero, such that the total number of objects in all piles doesn't exceed 2n.

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