1,1

Let the A133566 as a triangle = T. Then T * V, where V = any sequence as a vector with offset 1; gives a new sequence S such that S(2n) = V(2n) and S(2n-1) = T(n) + T(n-1). Example: T * [1,2,3...] = [1, 2, 5, 4, 9, 6, 13, 8, 17,...) = A114752. A133080 is identical to A133566 except that the subdiagonal = (1,0,1,0,...). A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, 9, 19,...).

Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,-1,0,0,0,0,0,0,...] DELTA [1,0,-2,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2007

As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (0,1,0,1,...) in the subdiagonal.) Triangle; odd rows, (n-2) zeros followed by 1, 1. Even rows, (n-1) zeros followed by 1.

Sum_{k, 0<=k<=n}T(n,k)=A040001(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 15 2007

First few rows of the triangle are:

1;

0, 1;

0, 1, 1;

0, 0, 0, 1;

0, 0, 0, 1, 1;

0, 0, 0, 0, 0, 1;

...

Cf. A133080, A114752.

Sequence in context: A102863 A131483 A077052 this_sequence A077051 A115955 A106344

Adjacent sequences: A133563 A133564 A133565 this_sequence A133567 A133568 A133569

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 16 2007