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%I #11 Jan 22 2023 09:16:26
%S 1,1,1,1,3,1,1,7,4,1,1,15,12,5,1,1,31,32,18,6,1,1,63,80,56,25,7,1,1,1,
%T 127,192,160,88,33,8,1,1,255,448,432,280,129,42,9,1
%N Triangle read by rows, derived from iterates of operations in which a current eigensequence becomes the left border of a new triangle; with triangles of the form: all 1's except the left border for triangles >1.
%C Row sums of the triangle = odd-indexed Fibonacci numbers, A001519(n).
%C Eigensequences of successive triangles tend to A001519: (1, 2, 5, 13, 34, ...).
%F The triangle is derived from operations in two parts. First, form an array in which rows = eigensequences of triangles of the form: (all 1's except the left border). First triangle has all 1's. Eigensequence of this triangle = (1, 2, 4, 8, 16, 32, ...). We shift this to the right = (1, 1, 2, 4, 8, 16, ...); creating a new triangle with this sequence as the left border. Eigensequence of this triangle = (1, 2, 5, 12, 28, 64, ...). We shift this to the right, making it the left border of the next triangle; and so on such that eigensequence of current triangle becomes next triangle's left border.
%F The second operation takes finite differences of the array. Reorienting the terms, we obtain the current triangle.
%e First few rows of the array:
%e 1, 2, 4, 8, 16, 32, ...
%e 1, 2, 5, 12, 28, 64, ...
%e 1, 2, 5, 13, 33, 82, ...
%e 1, 2, 5, 13, 34, 88, ...
%e 1, 2, 5, 13, 34, 89, ...
%e ...
%e Taking finite differences from the top, we reorient the terms obtaining triangle:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 4, 1;
%e 1, 15, 12, 5, 1;
%e 1, 31, 32, 18, 6, 1;
%e 1, 63, 80, 56, 25, 7, 1;
%e 1, 127, 192, 160, 88, 33, 8, 1;
%e 1, 255, 448, 432, 280, 129, 42, 9, 1;
%e ...
%Y Cf. A001519.
%K nonn,tabl,more
%O 1,5
%A _Gary W. Adamson_, Jul 25 2010
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