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

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;
  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;
  ...

Crossrefs

Cf. A000073.

Sequence in context: A208447 A320750 A117935 * A179749 A103462 A225641

Adjacent sequences: A224695 A224696 A224697 * A224699 A224700 A224701

Keyword

nonn,tabl

Author

Gary W. Adamson, Apr 15 2013

Status

approved