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

Author

Álvar Ibeas, Dec 16 2013

Keywords

• nonn

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.

Crossrefs

Cf. A000041, A048833, A050314.

Programs

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 *)

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).