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

1,1

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.

Links

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

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

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A303340 A115512 A115513 * A316917 A133985 A143062

Adjacent sequences: A133077 A133078 A133079 * A133081 A133082 A133083

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Sep 08 2007

Status

approved