A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 5, 0, 1, 0, 3, 0, 8, 1, 0, 2, 0, 5, 0, 13, 0, 1, 0, 3, 0, 8, 0, 21, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233

Offset

1,6

Comments

This triangle is different from A128619, which is A128619 = A127647 * A128174.

Links

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Formula

By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...

Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).

From G. C. Greubel, Mar 17 2024: (Start)

T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).

T(n, n) = A000045(n).

T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) * A355020(floor((n-1)/2)). (End)

Example

First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 3;
  1, 0, 2, 0, 5;
  0, 1, 0, 3, 0, 8;
  1, 0, 2, 0, 5, 0, 13;
  0, 1, 0, 3, 0, 8,  0, 21;
  1, 0, 2, 0, 5, 0, 13,  0, 34;
  0, 1, 0, 3, 0, 8,  0, 21,  0, 55;
  1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;
  ...

Mathematica

Table[Fibonacci[k]*Mod[n-k+1, 2], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)

Prog

(Magma) [((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024
(SageMath) flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Mar 17 2024

Crossrefs

Cf. A000045, A052952, A127647, A128174, A128619, A355020.

Sequence in context: A243978 A356898 A106844 * A363904 A125989 A125924

Adjacent sequences: A128615 A128616 A128617 * A128619 A128620 A128621

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 14 2007

Extensions

a(6) corrected and more terms from Georg Fischer, May 30 2023

Status

approved