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

1,1

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

Links

Table of n, a(n) for n=1..105.

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

Example

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;
  ...

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 *)

Crossrefs

Cf. A133080, A114752, A114753, A084938, A004001.

Sequence in context: A131483 A355655 A077052 * A185907 A321016 A077051

Adjacent sequences: A133563 A133564 A133565 * A133567 A133568 A133569

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Sep 16 2007

Extensions

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

Status

approved