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

0,3

Comments

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

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.

Formula

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

Example

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].

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 *)

Crossrefs

Cf. A050314.

Cf. A003987, A268390.

Sequence in context: A059749 A334213 A164142 * A204656 A070689 A339312

Adjacent sequences: A048830 A048831 A048832 * A048834 A048835 A048836

Keyword

nonn

Author

Christian G. Bower, Jun 15 1999

Status

approved