A094565 Triangle read by rows: binary products of Fibonacci numbers.

1, 2, 3, 5, 6, 8, 13, 15, 16, 21, 34, 39, 40, 42, 55, 89, 102, 104, 105, 110, 144, 233, 267, 272, 273, 275, 288, 377, 610, 699, 712, 714, 715, 720, 754, 987, 1597, 1830, 1864, 1869, 1870, 1872, 1885, 1974, 2584, 4181, 4791, 4880, 4893, 4895, 4896, 4901, 4935, 5168, 6765

Offset

1,2

Comments

Row n consists of n numbers, first F(2n-1) and last F(2n).

Central numbers: (1,6,40,273,...) = A081016.

Row sums: A001870.

Alternating row sums: 1,1,7,7,48,48,329,329; the sequence b=(1,7,48,329,...) is A004187, given by b(n)=F(4n+2)-b(n-1) for n>=2, with b(1)=1.

In each row, the difference between neighboring terms is a Fibonacci number.

Links

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

Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Formula

Row n: F(2)F(2n-1), F(4)F(2n-3), ..., F(2n)F(1).

Example

Triangle begins:
   1;
   2,   3;
   5,   6    8;
  13,  15,  16,  21;
  34,  39,  40,  42,  55;
  89, 102, 104, 105, 110, 144; ...

Mathematica

Table[Fibonacci[2*k]*Fibonacci[2*n-2*k+1], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 15 2019 *)

Prog

(PARI) row(n) = vector(n, k, fibonacci(2*k)*fibonacci(2*n-2*k+1));
tabl(nn) = for(n=1, nn, print(row(n))); \\ Michel Marcus, May 03 2016
(Magma) [Fibonacci(2*k)*Fibonacci(2*n-2*k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 15 2019
(Sage) [[fibonacci(2*k)*fibonacci(2*n-2*k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 15 2019
(GAP) Flat(List([1..12], n-> List([1..n], k-> Fibonacci(2*k)*Fibonacci(2*n-2*k+1) ))); # G. C. Greubel, Jul 15 2019

Crossrefs

Cf. A000045, A094566, A094568.

Sequence in context: A331865 A087360 A111501 * A364122 A034722 A144712

Adjacent sequences: A094562 A094563 A094564 * A094566 A094567 A094568

Keyword

nonn,tabl

Author

Clark Kimberling, May 12 2004

Status

approved