A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.

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

Offset: 1

Author

Gary W. Adamson, Apr 28 2007

Keywords

• nonn
• tabl

Comments

Alternate term operator, sums.

Let A129686 = matrix M, with V any sequence as a vector. Then M*V is the alternate term sum operator. Given V = [1,2,3,...], M*V = [1, 2, 4, 6, 8, 10, 12, 14, ...]. The analogous operation using A097807, (the pairwise operator), gives [1, 3, 5, 7, 9, 11, 13, 15, ...]. Binomial transform of A129686 = A124725. A129686 * A007318 = A129687. Row sums of A129686 = (1, 1, 2, 2, 2, ...).

Examples

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

Crossrefs

Cf. A124725, A129687.

Programs

Mathematica

T[n_, k_] := If[k == n || k == n-2, 1, 0];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019 *)

PARI

tabl(nn) = {t129686 = matrix(nn, nn, n, k, (k<=n)*((k==n) || (k==(n-2)))); for (n = 1, nn, for (k = 1, n, print1(t129686[n, k], ", ");););} \\ Michel Marcus, Feb 12 2014

Formula

As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal, (0,0,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal; with the rest zeros. (1, 0, 1, 0, 0, 0, ...) in every column.

Extensions

More terms from Michel Marcus, Feb 12 2014