A136157 Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.

3, 1, 3, 0, 1, 3, 0, 0, 1, 3, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3

Offset: 0

Author

Gary W. Adamson, Dec 16 2007

Keywords

• nonn
• tabl

Comments

Infinite lower triangular matrix with (3, 3, 3, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal, with the rest zeros.

Examples

			First few rows of the triangle:
  3;
  1, 3;
  0, 1, 3;
  0, 0, 1, 3;
  0, 0, 0, 1, 3;
  0, 0, 0, 0, 1, 3;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A136158.

Programs

Magma

function T(n,k) // T = A136157
  if k gt n-2 then return 2 + (-1)^(n+k);
  else return 0;
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023

Mathematica

Table[PadLeft[{1,3},n,{0}],{n,0,20}]//Flatten (* Harvey P. Dale, Apr 04 2018 *)

SageMath

def T(n,k): # T = A136157
    if k>n-2: return 2 + (-1)^(n+k)
    else: return 0
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Formula

From G. C. Greubel, Dec 26 2023: (Start)

T(n, k) = 3 if k = n, T(n, k) = 1 if k = n-1, otherwise T(n, k) = 0.

T(n, k) = 2 + (-1)^(n+k) for k >= n-1, otherwise T(n, k) = 0.

Sum_{k=0..n} T(n, k) = 4 - [n=0].

Sum_{k=0..n} (-1)^k*T(n, k) = (-2)^n + [n=0].

Sum_{k=0..floor(n/2)} T(n-k, k) = 2 + (-1)^n.

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (2 + (-1)^n)*(-1)^floor(n/2). (End)

Extensions

Offset changed by G. C. Greubel, Dec 26 2023