A238759 -id:A238759 - cp's OEIS frontend

A237711 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 is 2n.

1, 6, 7, 36, 13, 42, 43, 216, 49, 78, 55, 252, 85, 258, 259, 1296, 265, 294, 127, 468, 133, 330, 307, 1512, 337, 510, 343, 1548, 517, 1554, 1555, 7776, 1561, 1590, 559, 1764, 421, 762, 595, 2808, 601, 798, 463, 1980, 637, 1842, 1819, 9072, 1849

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

First differences of A237686.

			The P-positions with the total of 4 are permutations of (0,0,2,2) and (1,1,1,1). Therefore, a(2)=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. A237686 (partial sums), A048883 (3 piles), A238759 (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) = 6a(n), a(2n+2) = a(n+1) + a(n).

G.f.: Product_{k>=0} (1 + 6*x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Mar 16 2021

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.

Original entry on oeis.org

1, 11, 26, 126, 191, 341, 516, 1516, 2081, 2731, 3206, 4706, 5631, 7381, 9256, 19256, 24821, 30471, 33946, 40446, 44171, 48921, 52796, 67796, 76221, 85471, 91846, 109346, 119971, 138721, 158096, 258096, 313661, 369311

Offset: 0

Tanya Khovanova and Joshua Xiong, May 02 2014

nonn

Partial sums of A238759.

			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. Khovanova and J. Xiong, Nim Fractals, arXiv:1405.594291 [math.CO] (2014), p. 18 and J. Int. Seq. 17 (2014) # 14.7.8.

Cf. A238759 (first differences), A130665 (3 piles), A237686 (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], #[[5]] <= a &]], {a, 0, 35}]
(* Second program: *)
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]];
Array[a, 34, 0] (* Jean-François Alcover, Dec 14 2018 *)

a(2n+1) = 11a(n) + 5a(n-1), a(2n+2) = a(n+1) + 15a(n).

cp's OEIS Frontend

A237711 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 is 2n.

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