A166516 A product of consecutive doubled Fibonacci numbers.

1, 1, 2, 4, 10, 25, 65, 169, 442, 1156, 3026, 7921, 20737, 54289, 142130, 372100, 974170, 2550409, 6677057, 17480761, 45765226, 119814916, 313679522, 821223649, 2149991425, 5628750625, 14736260450, 38580030724, 101003831722

Offset: 0

Author

Paul Barry, Oct 16 2009

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Cf. A000045, A001654 (first differences), A166514.

Programs

Magma

A166516:= func< n | (n mod 2)*Fibonacci(n)^2 +((n+1) mod 2)*Fibonacci(n-1)*Fibonacci(n+1) >;
[A166516(n): n in [0..40]]; // G. C. Greubel, Aug 03 2024

Mathematica

CoefficientList[Series[(1-2x-x^2+x^3)/((1-x)(1+x)(1-3x+x^2)),{x,0,30}], x] (* or *) LinearRecurrence[{3,0,-3,1},{1,1,2,4},30] (* Harvey P. Dale, Dec 26 2013 *)

SageMath

f=fibonacci
def A166516(n): return (n%2)*f(n)^2 +((n+1)%2)*f(n-1)*f(n+1)
[A166516(n) for n in range(41)] # G. C. Greubel, Aug 03 2024

Formula

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

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

a(n) = Fibonacci(A166514(2*n))^2 + Fibonacci(A166514(2*n+1))^2.

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

a(n+1)*a(n+3) - a(n+2)^2 = Fibonacci(n+2)^2 * (1-(-1)^n)/2.

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4). - G. C. Greubel, May 15 2016