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

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• tabl

Comments

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

Examples

			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,
  ...
		

Links

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

Crossrefs

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

Cf. A128620 (row sums).

Programs

Magma

[((n+k+1) mod 2)*Fibonacci(n): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Mathematica

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

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)