cp's OEIS Frontend

A138134 a(n) = Sum_{i=0..n} Fibonacci(5*i).

%I A138134 #45 Feb 01 2019 05:33:07
%S A138134 0,5,60,670,7435,82460,914500,10141965,112476120,1247379290,
%T A138134 13833648315,153417510760,1701426266680,18869106444245,
%U A138134 209261597153380,2320746675131430,25737475023599115,285432971934721700,3165500166305537820
%N A138134 a(n) = Sum_{i=0..n} Fibonacci(5*i).
%C A138134 Partial sums of A102312.
%C A138134 Other sequences in the OEIS related to the sum of Fibonacci(k*n) (although not defined as such) are:
%C A138134   k = 1: A000071 = Fibonacci(n) - 1 (delete leading 0);
%C A138134   k = 2: A027941 = Fibonacci(2n+1) - 1;
%C A138134   k = 3: A099919 = (Fibonacci(3n+2) - 1)/2;
%C A138134   k = 4: A058038 = Fibonacci(2n)*Fibonacci(2n+2);
%C A138134   k = 6: A053606 = (Fibonacci(6n+3) - 2)/4.
%C A138134 These sequences appear to be second order linear inhomogeneous sequences of the form: a(0) = 0, a(1) = Fibonacci(k), a(n) = L(k)*a(n-1) + (-1)^(k+1)*a(n-2) + Fibonacci(k), where L(n) = A000032(n), n > 1.
%C A138134 The Koshy reference gives the closed form:
%C A138134   Sum_{i=0..n} Fibonacci(k*i) = (Fibonacci(n*k+k) - (-1)^k*Fibonacci(n*k) - Fibonacci(k))/(L(k) - (-1)^k - 1).
%D A138134 Thomas Koshy; Fibonacci and Lucas numbers with applications, Wiley,2001, p. 86.
%H A138134 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-10,-1).
%F A138134 G.f.: 5*x/((x - 1)*(x^2 + 11*x - 1)). - _R. J. Mathar_, Dec 09 2010
%F A138134 a(n) = 11*a(n) + a(n-1) + 5, n > 1.
%F A138134 a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3), n > 2.
%F A138134 a(n) = 1/11*(Fibonacci(5*n+5) + Fibonacci(5n) - 5).
%p A138134 with(combinat):fs5:=n-> sum(fibonacci(5*k),k=0..n):
%p A138134 seq(fs5(n),n=0..18)
%o A138134 (PARI) a(n)=(fibonacci(5*n+5)+fibonacci(5*n)-5)/11 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y A138134 Cf. A000071, A027941, A099919, A058038, A102312, A053606.
%Y A138134 Cf. also A000032, A000045.
%K A138134 nonn,easy
%O A138134 0,2
%A A138134 _Gary Detlefs_, Dec 07 2010