A133080 Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.

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

Offset: 1

Author

Gary W. Adamson, Sep 08 2007

Keywords

• nonn
• easy
• tabl

Comments

A133080 * [1,2,3,...] = A114753: (1, 3, 3, 7, 5, 11, 7, 15, ...).

Inverse of A133080: subdiagonal changes to (-1, 0, -1, 0, -1, ...); main diagonal unchanged.

A133080^(-1) * [1,2,3,...] = A093178: (1, 1, 3, 1, 5, 1, 7, 1, 9, ...).

In A133081, diagonal terms are switched with subdiagonal terms.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000034 (row sums), A114753, A093178, A133081.

Programs

Maple

A133080 := proc(n,k)
    if n = k then
        1;
    elif  k=n-1 and type(n,even) then
        1;
    else
        0 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2015

Mathematica

T[n_, k_] := If[k == n, 1, If[k == n - 1, (1 + (-1)^n)/2 , 0]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)

PARI

T(n, k) = if (k==n, 1, if (k == (n-1), 1 - (n % 2), 0)); \\ Michel Marcus, Feb 13 2014

PARI

firstrows(n) = {my(res = vector(binomial(n + 1, 2)), t=0); for(i=1, n, t+=i; res[t] = 1; if(i%2==0, res[t-1]=1)) ;res} \\ David A. Corneth, Oct 21 2017

Formula

Infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (1,0,1,0,1,...) in the subdiagonal.

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

T(n,n)=1. T(n,k)=0 if 1 <= k < n-1. T(n,n-1)=1 if n even. T(n,n-1)=0 if n odd. - R. J. Mathar, Feb 14 2015