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a(n) = Fibonacci(5n)/Fibonacci(n).
2

%I #39 Sep 08 2022 08:45:16

%S 5,55,305,2255,15005,104005,709805,4873055,33379505,228841255,

%T 1568358005,10750060805,73681030805,505019869255,3461450947505,

%U 23725155368255,162614587921805,1114577087604805,7639424691459005,52361396626646255,358890349406803505

%N a(n) = Fibonacci(5n)/Fibonacci(n).

%H Colin Barker, <a href="/A103326/b103326.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,15,-15,-5,1).

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

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

%F a(n) = 25*(Fibonacci(n)^4 + (-1)^n*Fibonacci(n)^2) + 5. - _Gary Detlefs_, Dec 22 2012

%F 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

%F a(n) = 5*A088545(n). - _Joerg Arndt_, Jul 16 2013

%F exp(Sum_{n >= 1} a(n)*x^n/n) = Sum_{n >= 0} A001656(n)*x^n. - _Peter Bala_, Mar 30 2015

%F 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

%p p:= (1+5^(1/2))/2: q:=(1-5^(1/2))/2:

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

%o (Magma) [Fibonacci(5*n)/Fibonacci(n): n in [1..50]]; // _Vincenzo Librandi_, Apr 20 2011

%o (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

%Y Fourth row of array A028412.

%Y Cf. A000032, A000045, A001656, A088545.

%K nonn,easy

%O 1,1

%A _Ralf Stephan_, Feb 03 2005

%E More terms from _Colin Barker_, Jul 16 2013