A334596 Number of values in A334556 with binary length n.

2, 0, 0, 2, 0, 2, 4, 2, 0, 8, 4, 8, 16, 8, 16, 32, 0, 32, 64, 32, 64, 128, 64, 128, 256, 128, 256, 512, 256, 512, 1024, 512, 0, 2048, 1024, 2048, 4096, 2048, 4096, 8192, 4096, 8192, 16384, 8192, 16384, 32768, 16384, 32768, 65536, 32768, 65536, 131072, 65536, 131072, 262144, 131072

Offset: 1

Author

Peter Kagey, May 07 2020

Keywords

• nonn
• base

Comments

All nonzero values are powers of two.

Examples

			For n = 11, the a(11) = 4 XOR-triangles of side length 11 are:
  1 0 1 0 1 1 0 0 0 1 1, 1 0 1 1 1 0 0 1 0 1 1,
   1 1 1 1 0 1 0 0 1 0    1 1 0 0 1 0 1 1 1 0
    0 0 0 1 1 1 0 1 1      0 1 0 1 1 1 0 0 1
     0 0 1 0 0 1 1 0        1 1 1 0 0 1 0 1
      0 1 1 0 1 0 1          0 0 1 0 1 1 1
       1 0 1 1 1 1            0 1 1 1 0 0
        1 1 0 0 0              1 0 0 1 0
         0 1 0 0                1 0 1 1
          1 1 0                  1 1 0
           0 1                    0 1
            1                      1
and their reflections across a vertical line.
By reading the first rows in binary, these XOR-triangles correspond to A334556(20) = 1379, A334556(21) = 1483, A334556(22) = 1589, and A334556(23) = 1693 respectively.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..1024
• MathOverflow user DSM, Number triangle
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A008611, A334556, A334591, A334592, A334593, A334594, A334595.

Programs

Mathematica

coeff[i_, j_, n_] := Binomial[i, j] - If[j + i == n, 1, 0];
A334596[1] = 2;
A334596[n_] := (
   nullsp = NullSpace[
     Table[coeff[i, j, n - 1], {i, 0, n - 1}, {j, 0, n - 1}],
     Modulus -> 2];
   If[AnyTrue[nullsp, #[[1]] == 1 &], 2^(Length[nullsp] - 1), 0]
   );

Formula

Conjectured formula:

a(1) = 2,

a(n) = 0 if n = 2^k + 1 for some k, and

a(n) = 2^A008611(n-4) otherwise.

Extensions

More terms from Rémy Sigrist, May 08 2020