A233810 -id:A233810 - cp's OEIS frontend

A372871 Number of compositions of n into n nonnegative parts such that their xor-sum is not zero.

0, 1, 2, 10, 28, 126, 236, 1716, 4376, 24310, 71452, 352716, 1036432, 5200300, 15661088, 77558760, 234338224, 1166803110, 3538500140, 17672631900, 53754680928, 269128937220, 811847006192, 4116715363800, 12392037943040, 63205303218876, 190668639444376

Offset: 0

Anna Ledworowska, May 15 2024

nonn

Number of starting configurations of Nim such that the 1st player wins, and the configurations are of the form {x_1, x_2, ..., x_n}, where x_i is the number of pieces on i-th stack (x_i>=0), and the sum of all pieces is n.

			For n=2 the a(2)=2 solutions are:  {0,2}, {2,0}.
For n=3 the a(3)=10 solutions are: {0,0,3}, {0,1,2}, {0,2,1}, {0,3,0}, {1,0,2}, {1,1,1}, {1,2,0}, {2,0,1}, {2,1,0}, {3,0,0}.

Alois P. Heinz, Table of n, a(n) for n = 0..350
C. L. Bouton, Nim, A Game with a Complete Mathematical Theory, Annals of Mathematics, Second Series, vol. 3 (1/4), 1902, 35-39.

Cf. A048833, A088218, A233810.

b:= proc(n, i, t) option remember; `if`(n=0, signum(t),
      add(b(n-j, i-1, Bits[Xor](j, t)), j=`if`(i=1, n, 0..n)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..32);  # Alois P. Heinz, May 15 2024

b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], Sum[b[n-j, i-1, BitXor[j, t]], {j, If[i == 1, n, 0], n}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jul 07 2025, after Alois P. Heinz *)

a(n) = A088218(n) if n is odd.

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

Original entry on oeis.org

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

A372871 Number of compositions of n into n nonnegative parts such that their xor-sum is not zero.

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