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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..81.

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 A226781

Adjacent sequences:  A233807 A233808 A233809 * A233811 A233812 A233813

KEYWORD

nonn

AUTHOR

Álvar Ibeas, Dec 16 2013

STATUS

approved

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Last modified October 2 08:34 EDT 2022. Contains 357191 sequences. (Running on oeis4.)