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.

1, 1, 2, 4, 6, 10, 16, 31, 43, 68, 98, 153, 213, 317, 443, 704, 971, 1415, 1975, 2818, 3865, 5401, 7366, 10142, 13639, 18438, 24583, 32861, 43345, 57268, 75175, 99119, 129278, 168796, 219614, 284887, 368546, 475919, 614379, 788845, 1012117, 1293980, 1654090

Offset: 0

Author

Christian G. Bower, Jun 15 1999

Keywords

• nonn

Comments

Number of different prime signatures of the 2n-almost primes in A268390. - Peter Munn, Dec 02 2021

Examples

			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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..750
• C. L. Bouton, Nim, a game with a complete mathematical theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.
• R. J. Nowakowski, G. Renault, E. Lamoureux, S. Mellon and T. Miller, The Game of timber!, hal-00985731, 2013.

Crossrefs

Cf. A050314.

Cf. A003987, A268390.

Programs

Maple

read("transforms") : # defines XORnos
A048833 := proc(n)
    local p, xrs,i,a ;
    if n = 0 then
        return 1 ;
    end if;
    a := 0 ;
    for p in combinat[partition](2*n) do
        xrs := op(1,p) ;
        for i from 2 to nops(p) do
            xrs := XORnos(xrs,op(i,p)) ;
        end do:
        if xrs = 0 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 29 2022

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}]]];
a[n_] := Coefficient[b[2n, 2n, 0], x, 0];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 25 2024, after Alois P. Heinz in A050314 *)

Formula

a(n) = A050314(2n, 0): column 0 of triangle.