cp's OEIS Frontend

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

%I A134310 #13 May 04 2022 01:18:09
%S A134310 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,
%T A134310 63,64,64,65,67,71,79,95,127,128,128,129,131,135,143,159,191,255,256,
%U A134310 256,257,259,263,271,287,319,383,511
%N A134310 (A000012 * A134309 + A134309 * A000012) - A000012, where the sequences are interpreted as lower triangular matrices.
%C A134310 From _M. F. Hasler_, Mar 29 2022: (Start)
%C A134310 Both A000012 and A134309 have offset 0, so this triangular matrix also has row and column indices starting at 0.
%C A134310 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:
%C A134310            ( 1         )   ( 1         )   ( 1         )
%C A134310            ( 1 1       )   ( 1 1       )   ( 1 1       )
%C A134310   (this) = ( 1 1 2     ) + ( 2 2 2     ) - ( 1 1 1     ) .  (End)
%C A134310            ( 1 1 2 4   )   ( 4 4 4 4   )   ( 1 1 1 1   )
%C A134310            ( 1 1 2 4 8 )   ( 8 8 8 8 8 )   ( 1 1 1 1 1 )
%F A134310 (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).
%F A134310 Row sums: A134311 = (1, 2, 7, 20, 51, 122, 281, 632, ...).
%e A134310 First few rows of the triangle:
%e A134310    1;
%e A134310    1,  1;
%e A134310    2,  2,  3;
%e A134310    4,  4,  5,  7;
%e A134310    8,  8,  9, 11, 15;
%e A134310   16, 16, 17, 19, 23, 31;
%e A134310   32, 32, 33, 35, 39, 47, 63;
%e A134310   ...
%o A134310 (PARI) A134310(r,c)=if(r>=c, 2^max(c-1,0)+2^max(r-1,0)-1)
%o A134310 matrix(8,8,i,j,A134310(i-1,j-1)) \\ _M. F. Hasler_, Mar 29 2022
%Y A134310 Cf. A000012 (all 1's), A134309 = diag(A011782 = 2^max(n-1,0), n >= 0), A000079.
%Y A134310 Row sums are A134311.
%K A134310 nonn,tabl
%O A134310 0,4
%A A134310 _Gary W. Adamson_, Oct 19 2007
%E A134310 Edited and offset corrected to 0 by _M. F. Hasler_, Mar 29 2022