A233810 Number of starting configurations of Nim with n pieces such that 1st player wins. Partitions of n such that their xor-sum is nonzero.

0, 1, 1, 3, 3, 7, 7, 15, 16, 30, 32, 56, 61, 101, 104, 176, 188, 297, 317, 490, 529, 792, 849, 1255, 1362, 1958, 2119, 3010, 3275, 4565, 4900, 6842, 7378, 10143, 10895, 14883, 16002, 21637, 23197, 31185, 33473, 44583, 47773, 63261, 67809, 89134, 95416, 124754, 133634, 173525, 185788, 239943, 257006, 329931, 353294, 451276, 483478, 614154, 657952, 831820, 891292, 1121505, 1201037, 1505499, 1612352, 2012558, 2154724, 2679689, 2868121, 3554345, 3803081, 4697205, 5024237, 6185689, 6613581, 8118264, 8674712, 10619863, 11343319, 13848650, 14784359, 18004327

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1500

C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.

Formula

a(n) = Sum_{k>0} A050314(n,k). [Row sums of A050314 minus the leftmost term on each row]

a(2n+1) = A000041(2n+1), a(2n) = A000041(2n)-A048833(n).

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, x^k, If[i<1, 0, Sum[b[n-i*j, i-1, If[EvenQ[j], k, BitXor[i, k]]], {j, 0, n/i}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]]; a[n_] := Total[Rest[T[n]]]; Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Nov 14 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A048833, A050314.

Sequence in context: A146035 A147190 A146450 * A263869 A174583 A370202

Adjacent sequences: A233807 A233808 A233809 * A233811 A233812 A233813

Keyword

nonn

Author

Álvar Ibeas, Dec 16 2013

Status

approved