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
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).
Programs
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Magma
[Fibonacci(5*n)/Fibonacci(n): n in [1..50]]; // Vincenzo Librandi, Apr 20 2011
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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);
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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
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
Extensions
More terms from Colin Barker, Jul 16 2013
Comments