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

0,1

Comments

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

Links

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

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)

Example

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;
  ...

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A136158.

Sequence in context: A167274 A201679 A131088 * A266260 A143353 A127172

Adjacent sequences: A136154 A136155 A136156 * A136158 A136159 A136160

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 16 2007

Extensions

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

Status

approved