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

Author

Clark Kimberling, May 12 2004

Keywords

• nonn
• tabl

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.

Examples

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

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.

Crossrefs

Cf. A000045, A094566, A094568.

Programs

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

Magma

[Fibonacci(2*k)*Fibonacci(2*n-2*k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 15 2019

Mathematica

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

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

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

Formula

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