A103326 a(n) = Fibonacci(5n)/Fibonacci(n).

5, 55, 305, 2255, 15005, 104005, 709805, 4873055, 33379505, 228841255, 1568358005, 10750060805, 73681030805, 505019869255, 3461450947505, 23725155368255, 162614587921805, 1114577087604805, 7639424691459005, 52361396626646255, 358890349406803505

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Formula

a(n) = L(4n) + (-1)^n*L(2n) + 1, where L(n) = A000032, the Lucas numbers.

a(n) = 1 + L(n)*L(3n). - Neven Juric, Jan 05 2009

a(n) = 25*(Fibonacci(n)^4 + (-1)^n*Fibonacci(n)^2) + 5. - Gary Detlefs, Dec 22 2012

G.f.: -5*x*(x^4 - 4*x^3 - 9*x^2 + 6*x + 1) /((x - 1)*(x^2 - 7*x + 1)*(x^2 + 3*x + 1)). - Colin Barker, Jul 16 2013

a(n) = 5*A088545(n). - Joerg Arndt, Jul 16 2013

exp(Sum_{n >= 1} a(n)*x^n/n) = Sum_{n >= 0} A001656(n)*x^n. - Peter Bala, Mar 30 2015

a(n) = 1 + (1/2*(7 - 3*sqrt(5)))^n + (1/2*(-3 - sqrt(5)))^n + (1/2*(-3 + sqrt(5)))^n + (1/2*(7 + 3*sqrt(5)))^n. - Colin Barker, Jun 03 2016

Maple

p:= (1+5^(1/2))/2: q:=(1-5^(1/2))/2:
seq(simplify(q^(4*n)+(p-2)^n+(q-2)^n+(3*p+2)^n+(-1)^(2*n)/4+3/4), n=1..19);

Prog

(Magma) [Fibonacci(5*n)/Fibonacci(n): n in [1..50]]; // Vincenzo Librandi, Apr 20 2011
(PARI) Vec(-5*x*(x^4-4*x^3-9*x^2+6*x+1)/((x-1)*(x^2-7*x+1)*(x^2+3*x+1)) + O(x^30)) \\ Colin Barker, Jun 03 2016

Crossrefs

Fourth row of array A028412.

Cf. A000032, A000045, A001656, A088545.

Sequence in context: A057722 A078216 A014700 * A060558 A348065 A014852

Adjacent sequences: A103323 A103324 A103325 * A103327 A103328 A103329

Keyword

nonn,easy

Author

Ralf Stephan, Feb 03 2005

Extensions

More terms from Colin Barker, Jul 16 2013

Status

approved