A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 8, 9, 11, 15, 16, 16, 17, 19, 23, 31, 32, 32, 33, 35, 39, 47, 63, 64, 64, 65, 67, 71, 79, 95, 127, 128, 128, 129, 131, 135, 143, 159, 191, 255, 256, 256, 257, 259, 263, 271, 287, 319, 383, 511

Offset

0,4

Comments

From M. F. Hasler, Mar 29 2022: (Start)

Both A000012 and A134309 have offset 0, so this triangular matrix also has row and column indices starting at 0.

Right (resp. left) multiplication by a diagonal matrix (such as A134309) amounts to multiplying the columns (resp. rows) of the other matrix by the diagonal elements. Therefore this matrix is the sum of the two lower triangular matrices with columns (resp. rows) filled with the same element given by sequence A134309 = (1, 1, 2, 4, 8, 16, ...), i. e., restricted to upper left 5 X 5 square:

( 1 ) ( 1 ) ( 1 )

( 1 1 ) ( 1 1 ) ( 1 1 )

(this) = ( 1 1 2 ) + ( 2 2 2 ) - ( 1 1 1 ) . (End)

( 1 1 2 4 ) ( 4 4 4 4 ) ( 1 1 1 1 )

( 1 1 2 4 8 ) ( 8 8 8 8 8 ) ( 1 1 1 1 1 )

Links

Table of n, a(n) for n=0..54.

Formula

(A000012 * A134309 + A134309 * A000012) - A000012, as infinite lower triangular matrices, where A000012 = (1; 1,1; 1,1,1; ...), and A134309 = diag(1, 1, 2, 4, 8, 16, ...) = diag(A011782 = 1 followed by 1, 2, 4, 8, ... = powers of 2).

Row sums: A134311 = (1, 2, 7, 20, 51, 122, 281, 632, ...).

Example

First few rows of the triangle:
   1;
   1,  1;
   2,  2,  3;
   4,  4,  5,  7;
   8,  8,  9, 11, 15;
  16, 16, 17, 19, 23, 31;
  32, 32, 33, 35, 39, 47, 63;
  ...

Prog

(PARI) A134310(r, c)=if(r>=c, 2^max(c-1, 0)+2^max(r-1, 0)-1)
matrix(8, 8, i, j, A134310(i-1, j-1)) \\ M. F. Hasler, Mar 29 2022

Crossrefs

Cf. A000012 (all 1's), A134309 = diag(A011782 = 2^max(n-1,0), n >= 0), A000079.

Row sums are A134311.

Sequence in context: A259200 A153155 A225085 * A308663 A305713 A259201

Adjacent sequences: A134307 A134308 A134309 * A134311 A134312 A134313

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 19 2007

Extensions

Edited and offset corrected to 0 by M. F. Hasler, Mar 29 2022

Status

approved