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%I #10 Dec 14 2023 08:56:56
%S 1,2,0,3,1,0,3,2,0,0,3,3,2,0,0,3,3,4,0,0,0,3,3,6,1,0,0,0,3,3,6,2,0,0,
%T 0,0,3,3,6,3,3,0,0,0,0,3,3,6,3,6,0,0,0,0,0,3,3,6,3,9,2,0,0,0,0,0,3,3,
%U 6,3,9,4,0,0,0,0,0,0,3,3,6,3,9,6,3
%N Triangle, row sums = A007729; derived from the generator for A002487, Stern's diatomic series.
%C Rows apparently tend to 3 * nonzero terms of Stern's diatomic series; i.e.,
%C 3 * (1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5,...) = (3, 3, 6, 3, 9, 6, 9, 3, 12,...)
%C Row sums = A007729: (1, 2, 4, 5, 8, 10, 13, 14, ...)
%F Triangle read by rows, Q*R*S; where Q = an infinite lower triangular matrix with all 1's, R = the generator for A002487, and S = a diagonalized variant of A002487 (nonzero terms of A002487 as the right diagonal and the rest zeros). R, the generator for A002487 is an irregular lower triangular matrix with (1, 1, 1, 0, 0, 0,...) in each column; but each successive column for k>0 is shifted down twice from the previous column.
%e First few rows of the triangle =
%e 1;
%e 2, 0;
%e 3, 1, 0;
%e 3, 2, 0, 0;
%e 3, 3, 2, 0, 0;
%e 3, 3, 4, 0, 0, 0;
%e 3, 3, 6, 1, 0, 0, 0;
%e 3, 3, 6, 2, 0, 0, 0, 0;
%e 3, 3, 6, 3, 3, 0, 0, 0, 0;
%e 3, 3, 6, 3, 6, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 2, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 4, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 3, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 6, 0, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 9, 1, 0, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 9, 2, 0, 0, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 9, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 9, 3, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0;
%e 3, 3, 6, 3, 9, 6, 9, 3, 12, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0;
%e ...
%Y Cf. A002487, A007729.
%K nonn,tabl,uned
%O 0,2
%A _Gary W. Adamson_, May 08 2010
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