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 A138134 a(n) = Sum_{i=0..n} Fibonacci(5*i). 7
 0, 5, 60, 670, 7435, 82460, 914500, 10141965, 112476120, 1247379290, 13833648315, 153417510760, 1701426266680, 18869106444245, 209261597153380, 2320746675131430, 25737475023599115, 285432971934721700, 3165500166305537820 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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). 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). MAPLE with(combinat):fs5:=n-> sum(fibonacci(5*k), k=0..n): seq(fs5(n), n=0..18) PROG (PARI) a(n)=(fibonacci(5*n+5)+fibonacci(5*n)-5)/11 \\ Charles R Greathouse IV, Jun 11 2015 CROSSREFS Cf. A000071, A027941, A099919, A058038, A102312, A053606. Cf. also A000032, A000045. Sequence in context: A212700 A099672 A320361 * A283779 A093885 A192948 Adjacent sequences:  A138131 A138132 A138133 * A138135 A138136 A138137 KEYWORD nonn,easy AUTHOR Gary Detlefs, Dec 07 2010 STATUS approved

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Last modified October 21 06:01 EDT 2019. Contains 328291 sequences. (Running on oeis4.)