A140068 Triangle read by rows, n-th row = (n-1)-th power of the matrix X * [1,0,0,0,...] where X = an infinite lower triangular matrix with [1,2,1,2,1,2,...] in the main diagonal and [1,1,1,...] in the subdiagonal.

1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 11, 6, 1, 1, 31, 26, 23, 7, 1, 1, 63, 57, 72, 30, 9, 1, 1, 127, 120, 201, 102, 48, 10, 1, 1, 255, 247, 522, 303, 198, 58, 12, 1, 1, 511, 502, 1291, 825, 699, 256, 82, 13, 1, 1, 1023, 1013, 3084, 2116, 2223, 955, 420, 95, 15, 1, 1, 2047, 2036

Offset: 1

Author

Gary W. Adamson and Roger L. Bagula, May 04 2008

Keywords

• nonn
• tabl

Comments

Sum of n-th row terms = odd-indexed Fibonacci numbers, F(2n+1); e.g. sum of row 5 terms = (1 + 15 + 11 + 6 + 1) = 34 = F(9).

The triangle is a companion to A140069 (having row sums = even-indexed Fibonacci numbers).

Examples

			First few rows of the triangle are:
  1;
  1,   1;
  1,   3,   1;
  1,   7,   4,   1;
  1,  15,  11,   6,   1;
  1,  31,  26,  23,   7,   1;
  1,  63,  57,  72,  30,   9,   1;
  1, 127, 120, 201, 102,  48,  10,   1;
  1, 255, 247, 522, 303, 198,  58,  12,   1;
  ...
		

Crossrefs

Cf. A140069.

Formula

Triangle read by rows, n-th row = (n-1)-th power of the matrix X * [1,0,0,0,...] where X = an infinite lower triangular matrix with [1,2,1,2,1,2,...] in the main diagonal and [1,1,1,...] in the subdiagonal. Given the matrix X, perform X * [1,0,0,0,...] and then iterate: X * (result), etc. and record the result as each successive row of the triangle.