A372871 -id:A372871 - cp's OEIS frontend

A372967 Number of compositions of 2n into 2n nonnegative parts such that their xor-sum is zero.

1, 1, 7, 226, 2059, 20926, 315646, 4397212, 66201971, 999067510, 15168583482, 240202475668, 3731763898510, 57290627029676, 887861389544668, 13713341876387776, 210889953761225667, 3248614469788303782, 50091681144815341810, 772966100038376636332

Offset: 0

Anna Ledworowska, May 18 2024

nonn

Number of starting configurations of Nim with 2n pieces such that 2nd player wins, and the configurations are of the form {x_1, x_2, ..., x_2n}, where x_i is the number of pieces on i-th stack (x_i>=0), and the sum of all pieces is 2n.

			For n=1 the a(1)=1 solution is {1,1}.
For n=2 the a(2)=7 solutions are {0,0,2,2}, {0,2,0,2}, {0,2,2,0}, {1,1,1,1}, {2,0,0,2}, {2,0,2,0}, {2,2,0,0}.

C. L. Bouton, Nim, A Game with a Complete Mathematical Theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.

Cf. A088218, A131577, A372871, A048833, A233810.

b:= proc(n, i, t) option remember; `if`(n=0, 1-signum(t),
      add(b(n-j, i-1, Bits[Xor](j, t)), j=`if`(i=1, n, 0..n)))
    end:
a:= n-> b(2*n$2, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, May 22 2024

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1-Sign[t],
   Sum[b[n-j, i-1, BitXor[j, t]], {j, If[i == 1, {n}, Range[0, n]]}]];
a[n_] := b[2n, 2n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, May 30 2024, after Alois P. Heinz *)

a(n) = A088218(2n) - A372871(2n).

a(n) mod 2 = 1 <=> n in { A131577 }.

More terms from Alois P. Heinz, May 22 2024

cp's OEIS Frontend

A372967 Number of compositions of 2n into 2n nonnegative parts such that their xor-sum is zero.

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