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A334596 Number of values in A334556 with binary length n.

%I A334596 #20 May 07 2021 00:55:27
%S A334596 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,
%T A334596 256,512,256,512,1024,512,0,2048,1024,2048,4096,2048,4096,8192,4096,
%U A334596 8192,16384,8192,16384,32768,16384,32768,65536,32768,65536,131072,65536,131072,262144,131072
%N A334596 Number of values in A334556 with binary length n.
%C A334596 All nonzero values are powers of two.
%H A334596 Peter Kagey, <a href="/A334596/b334596.txt">Table of n, a(n) for n = 1..1024</a>
%H A334596 MathOverflow user DSM, <a href="https://mathoverflow.net/q/359138/104733">Number triangle</a>
%H A334596 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A334596 Conjectured formula:
%F A334596 a(1) = 2,
%F A334596 a(n) = 0 if n = 2^k + 1 for some k, and
%F A334596 a(n) = 2^A008611(n-4) otherwise.
%e A334596 For n = 11, the a(11) = 4 XOR-triangles of side length 11 are:
%e A334596   1 0 1 0 1 1 0 0 0 1 1, 1 0 1 1 1 0 0 1 0 1 1,
%e A334596    1 1 1 1 0 1 0 0 1 0    1 1 0 0 1 0 1 1 1 0
%e A334596     0 0 0 1 1 1 0 1 1      0 1 0 1 1 1 0 0 1
%e A334596      0 0 1 0 0 1 1 0        1 1 1 0 0 1 0 1
%e A334596       0 1 1 0 1 0 1          0 0 1 0 1 1 1
%e A334596        1 0 1 1 1 1            0 1 1 1 0 0
%e A334596         1 1 0 0 0              1 0 0 1 0
%e A334596          0 1 0 0                1 0 1 1
%e A334596           1 1 0                  1 1 0
%e A334596            0 1                    0 1
%e A334596             1                      1
%e A334596 and their reflections across a vertical line.
%e A334596 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.
%t A334596 coeff[i_, j_, n_] := Binomial[i, j] - If[j + i == n, 1, 0];
%t A334596 A334596[1] = 2;
%t A334596 A334596[n_] := (
%t A334596    nullsp = NullSpace[
%t A334596      Table[coeff[i, j, n - 1], {i, 0, n - 1}, {j, 0, n - 1}],
%t A334596      Modulus -> 2];
%t A334596    If[AnyTrue[nullsp, #[[1]] == 1 &], 2^(Length[nullsp] - 1), 0]
%t A334596    );
%Y A334596 Cf. A008611, A334556, A334591, A334592, A334593, A334594, A334595.
%K A334596 nonn,base
%O A334596 1,1
%A A334596 _Peter Kagey_, May 07 2020
%E A334596 More terms from _Rémy Sigrist_, May 08 2020