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

1,1

Comments

All nonzero values are powers of two.

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

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.

Example

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.

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]
   );

Crossrefs

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

Sequence in context: A262938 A143068 A261202 * A291900 A263146 A365047

Adjacent sequences: A334593 A334594 A334595 * A334597 A334598 A334599

Keyword

nonn,base

Author

Peter Kagey, May 07 2020

Extensions

More terms from Rémy Sigrist, May 08 2020

Status

approved