A224698 - OEIS

%I #15 Feb 08 2022 22:14:47

%S 1,1,1,1,2,1,1,2,3,1,1,2,5,4,1,1,2,6,9,5,1,1,2,6,14,14,6,1,1,2,6,17,

%T 27,20,7,1,1,2,6,18,40,46,27,8,1,1,2,6,18,49,81,72,35,9,1,1,2,6,18,53,

%U 120,143,106,44,10,1

%N Tribonacci triangle by rows, generated from (1, 1, 1, 0, 0, 0, ...).

%C Row sums = the tribonacci sequence, A000073; starting (1, 2, 4, 7, 13, 24, ...).

%C The latter sequence is the INVERT transform of (1, 1, 1, 0, 0, 0, ...).

%F Given G = an infinite Toeplitz triangle with (1, 1, 1, 0, 0, 0, ...) in every column and row 1 = vector (1, 1, 1, 1, 1, 1, ...). Perform the operation G * V, then prepend the result with "1" to create the next V. Repeat the operation, forming an array. Take finite differences of the array columns which become rows of the triangle.

%e First few rows of the array are:

%e   1, 1, 1, 1, 1,  1,  1,  1, ...

%e   1, 1, 2, 3, 3,  3,  3,  3, ...

%e   1, 1, 2, 4, 6,  8,  9,  9, ...

%e   1, 1, 2, 4, 7, 12, 18, 23, ...

%e   1, 1, 2, 4, 7, 12, 23, 37, ...

%e   ...

%e Deleting the first column, we take finite differences of remaining columns such that the column (1, 3, 6, 7, 7, ...) becomes the row with terms (1, 2, 3, 1, 0, 0, 0, ...).

%e First few rows of the triangle:

%e   1;

%e   1, 1;

%e   1, 2, 1;

%e   1, 2, 3,  1;

%e   1, 2, 5,  4,  1;

%e   1, 2, 6,  9,  5,   1;

%e   1, 2, 6, 14, 14,   6,   1;

%e   1, 2, 6, 17, 27,  20,   7,   1;

%e   1, 2, 6, 18, 40,  46,  27,   8,  1;

%e   1, 2, 6, 18, 49,  81,  72,  35,  9,  1;

%e   1, 2, 6, 18, 53, 120, 143, 106, 44, 10, 1;

%e   ...

%Y Cf. A000073.

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, Apr 15 2013