A226958 a(n) = Fibonacci(n-2)*Fibonacci(n)*Fibonacci(n+2).

2, 0, 10, 24, 130, 504, 2210, 9240, 39338, 166320, 705058, 2985840, 12649570, 53582256, 226981610, 961503816, 4073004770, 17253510120, 73087065922, 309601740360, 1311494081482, 5555577978720, 23533806138050, 99690802301664, 422297015715650, 1788878864564064, 7577812474943050

Offset: 1

Author

Ron Knott, Jun 27 2013

Keywords

• nonn

Examples

			a(3) = F(1)*F(3)*F(5) = 1*2*5 = 10.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Crossrefs

Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

Products of 3 Fibonaccis: A065563, A056570, A220362, A110224.

Programs

Mathematica

Table[Fibonacci[n - 2] Fibonacci[n] Fibonacci[n + 2], {n, 1, 20}]
LinearRecurrence[{3,6,-3,-1},{2,0,10,24},30] (* Harvey P. Dale, Apr 10 2022 *)
Join[{2},#[[1]]#[[3]]#[[5]]&/@Partition[Fibonacci[Range[0,40]],5,1]] (* Harvey P. Dale, May 20 2025 *)

PARI

a(n)=fibonacci(n-2)*fibonacci(n)*fibonacci(n+2); \\ Joerg Arndt, Jul 07 2013

Formula

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).

G.f.: 2*(1-3*x-x^2)/(1-3*x-6*x^2+3*x^3+x^4).

a(n) = Lucas(n-1)*Fibonacci(n+2) = Fibonacci(n-2)*Lucas(n+1).

a(n) = (1/5)*(Fibonacci(3*n)-8*(-1)^n*Fibonacci(n)). - Ehren Metcalfe, Mar 26 2016

For n >= 3, a(n) is the numerator of the continued fraction [1,..,1, 3 ,1,..,1, 3 ,1,..,1] with three runs of 1's each of length n-3 and each separated by a single 3. For example, a(5)=130 which is the numerator of the continued fraction [1,1, 3 ,1,1, 3 ,1,1]. - Greg Dresden, Jan 01 2022

Extensions

More terms from Joerg Arndt, Jul 07 2013