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A334596 Number of values in A334556 with binary length n. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All nonzero values are powers of two.
LINKS
MathOverflow user DSM, Number triangle
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
Sequence in context: A262938 A143068 A261202 * A291900 A263146 A365047
KEYWORD
nonn,base
AUTHOR
Peter Kagey, May 07 2020
EXTENSIONS
More terms from Rémy Sigrist, May 08 2020
STATUS
approved

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Last modified February 28 05:20 EST 2024. Contains 370379 sequences. (Running on oeis4.)