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A048833 Number of starting positions of Nim with 2n pieces such that 2nd player wins. Partitions of 2n such that xor-sum of partitions is 0.

%I A048833 #35 May 14 2024 11:40:06
%S A048833 1,1,2,4,6,10,16,31,43,68,98,153,213,317,443,704,971,1415,1975,2818,
%T A048833 3865,5401,7366,10142,13639,18438,24583,32861,43345,57268,75175,99119,
%U A048833 129278,168796,219614,284887,368546,475919,614379,788845,1012117,1293980,1654090
%N A048833 Number of starting positions of Nim with 2n pieces such that 2nd player wins. Partitions of 2n such that xor-sum of partitions is 0.
%C A048833 Number of different prime signatures of the 2n-almost primes in A268390. - _Peter Munn_, Dec 02 2021
%H A048833 Alois P. Heinz, <a href="/A048833/b048833.txt">Table of n, a(n) for n = 0..750</a>
%H A048833 C. L. Bouton, <a href="http://www.jstor.org/stable/1967631">Nim, a game with a complete mathematical theory</a>, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.
%H A048833 R. J. Nowakowski, G. Renault, E. Lamoureux, S. Mellon and T. Miller, <a href="https://hal.archives-ouvertes.fr/hal-00985731">The Game of timber!</a>,  hal-00985731, 2013.
%F A048833 a(n) = A050314(2n, 0): column 0 of triangle.
%e A048833 For n=4 the 6 partitions of 8 are [1, 1, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2, 2], [2, 2, 2, 2], [1, 1, 1, 2, 3], [1, 1, 3, 3] and [4, 4].
%p A048833 read("transforms") : # defines XORnos
%p A048833 A048833 := proc(n)
%p A048833     local p, xrs,i,a ;
%p A048833     if n = 0 then
%p A048833         return 1 ;
%p A048833     end if;
%p A048833     a := 0 ;
%p A048833     for p in combinat[partition](2*n) do
%p A048833         xrs := op(1,p) ;
%p A048833         for i from 2 to nops(p) do
%p A048833             xrs := XORnos(xrs,op(i,p)) ;
%p A048833         end do:
%p A048833         if xrs = 0 then
%p A048833             a := a+1 ;
%p A048833         end if;
%p A048833     end do:
%p A048833     a ;
%p A048833 end proc: # _R. J. Mathar_, Apr 29 2022
%t A048833 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 A048833 a[n_] := Coefficient[b[2n, 2n, 0], x, 0];
%t A048833 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Mar 25 2024, after _Alois P. Heinz_ in A050314 *)
%Y A048833 Cf. A050314.
%Y A048833 Cf. A003987, A268390.
%K A048833 nonn
%O A048833 0,3
%A A048833 _Christian G. Bower_, Jun 15 1999