%I #45 Feb 01 2019 05:33:07

%S 0,5,60,670,7435,82460,914500,10141965,112476120,1247379290,

%T 13833648315,153417510760,1701426266680,18869106444245,

%U 209261597153380,2320746675131430,25737475023599115,285432971934721700,3165500166305537820

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

%C Partial sums of A102312.

%C Other sequences in the OEIS related to the sum of Fibonacci(k*n) (although not defined as such) are:

%C k = 1: A000071 = Fibonacci(n) - 1 (delete leading 0);

%C k = 2: A027941 = Fibonacci(2n+1) - 1;

%C k = 3: A099919 = (Fibonacci(3n+2) - 1)/2;

%C k = 4: A058038 = Fibonacci(2n)*Fibonacci(2n+2);

%C k = 6: A053606 = (Fibonacci(6n+3) - 2)/4.

%C 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 The Koshy reference gives the closed form:

%C Sum_{i=0..n} Fibonacci(k*i) = (Fibonacci(n*k+k) - (-1)^k*Fibonacci(n*k) - Fibonacci(k))/(L(k) - (-1)^k - 1).

%D Thomas Koshy; Fibonacci and Lucas numbers with applications, Wiley,2001, p. 86.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (12,-10,-1).

%F G.f.: 5*x/((x - 1)*(x^2 + 11*x - 1)). - _R. J. Mathar_, Dec 09 2010

%F a(n) = 11*a(n) + a(n-1) + 5, n > 1.

%F a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3), n > 2.

%F a(n) = 1/11*(Fibonacci(5*n+5) + Fibonacci(5n) - 5).

%p with(combinat):fs5:=n-> sum(fibonacci(5*k),k=0..n):

%p seq(fs5(n),n=0..18)

%o (PARI) a(n)=(fibonacci(5*n+5)+fibonacci(5*n)-5)/11 \\ _Charles R Greathouse IV_, Jun 11 2015

%Y Cf. A000071, A027941, A099919, A058038, A102312, A053606.

%Y Cf. also A000032, A000045.

%K nonn,easy

%O 0,2

%A _Gary Detlefs_, Dec 07 2010