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A334596
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Number of values in A334556 with binary length n.
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6
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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
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OFFSET
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1,1
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COMMENTS
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All nonzero values are powers of two.
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LINKS
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FORMULA
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Conjectured formula:
a(1) = 2,
a(n) = 0 if n = 2^k + 1 for some k, and
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EXAMPLE
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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.
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MATHEMATICA
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coeff[i_, j_, n_] := Binomial[i, j] - If[j + i == n, 1, 0];
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]
);
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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