A102312 a(n) = Fibonacci(5*n).

0, 5, 55, 610, 6765, 75025, 832040, 9227465, 102334155, 1134903170, 12586269025, 139583862445, 1548008755920, 17167680177565, 190392490709135, 2111485077978050, 23416728348467685, 259695496911122585, 2880067194370816120, 31940434634990099905

Offset: 0

Author

Ralf Stephan, Jan 06 2005

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..950
• Michael D.Hirschhorn, A Naive Proof that F5n = 0 (mod 5), Fib. Q. 51(3), 2013, 256-258.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (11,1).

Crossrefs

Essentially the fifth column of array A102310.

Cf. A049666. [Zerinvary Lajos, May 15 2009]

Cf. A138134 (partial sums).

Programs

Magma

[Fibonacci(5*n): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Maple

seq(combinat:-fibonacci(5*n), n=0..100); # Robert Israel, Dec 12 2014

Mathematica

Table[ Fibonacci[5n], {n, 0, 17}] (* Robert G. Wilson v, Jan 09 2005 *)

PARI

vector(18,n,fibonacci(5*n)) \\ Edward Jiang, Dec 11 2014

PARI

concat(0, Vec(5*x/(1-11*x-x^2) + O(x^30))) \\ Colin Barker, Nov 10 2016

Sage

[fibonacci(5*n) for n in range(0, 18)] # Zerinvary Lajos, May 15 2009

Formula

G.f.: 5*x/(1-11*x-x^2).

a(n) = A000045(5*n) = 5*A049666(n).

a(n) = Fibonacci(2*n)*Lucas(3*n)+Fibonacci(n). Lucas =A000032(n), Fibonacci=A000045(n). - Gary Detlefs, Dec 22 2012

a(n) = (-((11 - 5*sqrt(5))/2)^n + ((11+5*sqrt(5))/2)^n)/sqrt(5). - Colin Barker, Nov 10 2016

a(n) = 11*a(n-1)+a(n-2). - Mike Speciner, Aug 20 2024