A224698 - OEIS

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A224698

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

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, 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, 120, 143, 106, 44, 10, 1

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OFFSET

1,5

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..66.

FORMULA

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.

EXAMPLE

First few rows of the array are:

1, 1, 1, 1, 1, 1, 1, 1, ...

1, 1, 2, 3, 3, 3, 3, 3, ...

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

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

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

...

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, ...).

First few rows of the triangle:

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, 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, 120, 143, 106, 44, 10, 1;

...