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

0, 5, 60, 670, 7435, 82460, 914500, 10141965, 112476120, 1247379290, 13833648315, 153417510760, 1701426266680, 18869106444245, 209261597153380, 2320746675131430, 25737475023599115, 285432971934721700, 3165500166305537820

Offset: 0

Author

Gary Detlefs, Dec 07 2010

Keywords

• nonn
• easy

Comments

Partial sums of A102312.

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

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

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

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

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

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

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.

The Koshy reference gives the closed form:

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

References

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

Links

• Index entries for linear recurrences with constant coefficients, signature (12,-10,-1).

Crossrefs

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

Cf. also A000032, A000045.

Programs

Maple

with(combinat):fs5:=n-> sum(fibonacci(5*k),k=0..n):
seq(fs5(n),n=0..18)

PARI

a(n)=(fibonacci(5*n+5)+fibonacci(5*n)-5)/11 \\ Charles R Greathouse IV, Jun 11 2015

Formula

G.f.: 5*x/((x - 1)*(x^2 + 11*x - 1)). - R. J. Mathar, Dec 09 2010

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

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

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