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Odious count triangle, T(n,k) = A010060(n) * A010060(k); 1 <= k <= n.
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%I #14 Aug 05 2023 13:16:33

%S 1,1,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1,1,1,0,1,0,0,

%T 1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1,1,0,0,1,0,0,

%U 0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,1,1,0,0,1,0,1

%N Odious count triangle, T(n,k) = A010060(n) * A010060(k); 1 <= k <= n.

%C Row sums = (1, 2, 0, 3, 0, 0, 4, 5, 0, 0, 0, 6, ...) = A102390, the Odious count sequence starting with offset 1.

%C Odious count triangle, T(n,k) = A010060(n) * A010060(k); 1 <= k <= n. If n is an odious number (1, 2, 4, 7, 8, ...), row n consists of the first n terms of the M-T sequence (A010060) starting with offset 1: (1, 1, 0, 1, 0, 0, 1, 1, ...); otherwise 0. Let X be an infinite lower triangular matrix with (1, 1, 0, 1, 0, 0, 1, ...) in the main diagonal and the rest zeros (i.e., A010060 * 0^(n-k), with offset 1). Then X * A000012 * X = triangle A143466.

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 0, 0, 0;

%e 1, 1, 0, 1;

%e 0, 0, 0, 0, 0;

%e 0, 0, 0, 0, 0, 0;

%e 1, 1, 0, 1, 0, 0, 1;

%e ...

%e T(7,3) = 1 since given the first few terms of the M-T sequence starting with offset 1, (1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, ...), product of 7th and 3rd terms = 1.

%t T[n_, k_] := ThueMorse[n] * ThueMorse[k]; Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Aug 05 2023 *)

%Y Cf. A000012, A010060, A102390.

%K nonn,tabl

%O 1,1

%A _Gary W. Adamson_, Aug 17 2008

%E More terms from _Amiram Eldar_, Aug 05 2023