A133566 Triangle read by rows: (1,1,1,...) on the main diagonal and (0,1,0,1,...) on the subdiagonal.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Gary W. Adamson, Sep 16 2007

Keywords

• nonn
• tabl
• easy

Comments

Usually regarded as a square matrix T when combined with other matrices and column vectors.

Then T * V, where V = any sequence regarded as a column vector with offset 1 is a new sequence S [called an interpolation transform] given by S(2n) = V(2n), S(2n-1) = V(2n) + V(2n-1). Example: If T * [1,2,3,...], S = [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 Deléham, Dec 15 2007

Examples

			First few rows of the triangle:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1;
  ...
		

Crossrefs

Cf. A133080, A114752, A114753, A084938, A040001.

Programs

Maple

A133566 := proc(n,k)
    if n = k then
        1;
    elif  k=n-1 and type(n,odd) then
        1;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

Mathematica

T[n_, k_] := Which[n == k, 1, k == n - 1 && OddQ[n], 1, True, 0];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 24 2023 *)

Formula

Odd rows: (n-2) zeros followed by 1, 1. Even rows: (n-1) zeros followed by 1.

Sum_{k=0..n} T(n,k) = A040001(n). - Philippe Deléham, Dec 15 2007

G.f.: (-1-x*y-x^2*y)*x*y/((-1+x*y)*(1+x*y)). - R. J. Mathar, Aug 11 2015

Extensions

Entry revised by N. J. A. Sloane, Jun 20 2015