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A134310
(A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.
2
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 )
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
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