A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

1, 0, 1, 2, 0, 2, 0, 3, 0, 3, 5, 0, 5, 0, 5, 0, 8, 0, 8, 0, 8, 13, 0, 13, 0, 13, 0, 13, 0, 21, 0, 21, 0, 21, 0, 21, 34, 0, 34, 0, 34, 0, 34, 0, 34, 0, 55, 0, 55, 0, 55, 0, 55, 0, 55

Offset

1,4

Comments

This triangle is different from A128618, which is equal to A128174 * A127647.

Links

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

Formula

T(n, k) = A127647 * A128174, an infinite lower triangular matrix. In odd rows, n terms of F(n), 0, F(n),...; in the n-th row. In even rows, n terms of 0, F(n), 0,...; in the n-th row.

Sum_{k=1..n} T(n, k) = A128620(n-1).

From G. C. Greubel, Mar 16 2024: (Start)

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

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^n*A128620(n-1).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A096140(floor((n + 1)/2)).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*( Fibonacci(n-1) + (-1)^floor((n-1)/2) * Fibonacci(floor((n-3)/2)) ). (End)

Example

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

Mathematica

Table[Fibonacci[n]*Mod[n+k+1, 2], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Mar 16 2024 *)

Prog

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

Crossrefs

Cf. A000045, A096140, A127646, A128174, A128610, A128618, A128620.

Cf. A128620 (row sums).

Sequence in context: A363824 A025805 A029192 * A008613 A165685 A035457

Adjacent sequences: A128616 A128617 A128618 * A128620 A128621 A128622

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 14 2007

Status

approved