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

0,3

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).

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

Mathematica

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

Prog

(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
(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
(Python)
from gmpy2 import fib2
def A166516(n):
    b, a = fib2(n+1&-2)
    return a*(a if n&1 else a+b) # Chai Wah Wu, Jul 14 2026

Crossrefs

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

Sequence in context: A026269 A000645 A005958 * A230552 A230555 A189912

Adjacent sequences: A166513 A166514 A166515 * A166517 A166518 A166519

Keyword

easy,nonn,changed

Author

Paul Barry, Oct 16 2009

Status

approved