A179745 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.

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, 127, 192, 160, 88, 33, 8, 1, 1, 255, 448, 432, 280, 129, 42, 9, 1

Offset: 1

Author

Gary W. Adamson, Jul 25 2010

Keywords

• nonn
• tabl
• more

Comments

Row sums of the triangle = odd-indexed Fibonacci numbers, A001519(n).

Eigensequences of successive triangles tend to A001519: (1, 2, 5, 13, 34, ...).

Examples

			First few rows of the array:
  1,  2,  4,  8, 16, 32, ...
  1,  2,  5, 12, 28, 64, ...
  1,  2,  5, 13, 33, 82, ...
  1,  2,  5, 13, 34, 88, ...
  1,  2,  5, 13, 34, 89, ...
  ...
Taking finite differences from the top, we reorient the terms obtaining triangle:
  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, 127, 192, 160,  88,  33,   8,   1;
  1, 255, 448, 432, 280, 129,  42,   9,   1;
  ...
		

Crossrefs

Cf. A001519.

Formula

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.

The second operation takes finite differences of the array. Reorienting the terms, we obtain the current triangle.

Extensions

Duplicate term removed by Stefano Spezia, Jul 16 2025