A143211 Triangle read by rows, T(n,k) = Fibonacci(n)*Fibonacci(k).

1, 1, 1, 2, 2, 4, 3, 3, 6, 9, 5, 5, 10, 15, 25, 8, 8, 16, 24, 40, 64, 13, 13, 26, 39, 65, 104, 169, 21, 21, 42, 63, 105, 168, 273, 441, 34, 34, 68, 102, 170, 272, 442, 714, 1156, 55, 55, 110, 165, 275, 440, 715, 1155, 1870, 3025, 89, 89, 178, 267, 445, 712, 1157, 1869

Offset: 1

Author

Gary W. Adamson, Jul 30 2008

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  4;
   3,  3,  6,  9;
   5,  5, 10, 15,  25;
   8,  8, 16, 24,  40,  64;
  13, 13, 26, 39,  65, 104, 169;
  21, 21, 42, 63, 105, 168, 273, 441;
  ...
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A000045 (left border), A007598 (right border), A127647,

Cf. A024458 (diagonal row sums), A143212 (row sums).

Programs

Magma

F:=Fibonacci; [F(n)*F(k): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 20 2024

Mathematica

With[{F=Fibonacci}, Table[F[k]*F[n], {n,12}, {k,n}]]//Flatten (* G. C. Greubel, Jul 20 2024 *)

SageMath

def A143211(n,k): return fibonacci(n)*fibonacci(k)
flatten([[A143211(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 20 2024

Formula

T(n, k) = Fibonacci(n)*Fibonacci(k).

T(n, k) = A127647 * A000012 * A127647, as infinite lower triangular matrices.

T(n, 1) = A000045(n).

T(n, n) = A007598(n).

Sum_{k=1..n} T(n, k) = A143212(n).

From G. C. Greubel, Jul 20 2024: (Start)

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

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A024458(n). (End)