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A138134
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a(n)=Sum(Fibonacci(5*k),k=0..n).
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7
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0, 5, 60, 670, 7435, 82460, 914500, 10141965, 112476120, 1247379290, 13833648315, 153417510760, 1701426266680, 18869106444245, 209261597153380, 2320746675131430, 25737475023599115, 285432971934721700, 3165500166305537820
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Partial sums of A102312.
Other sequences in the OEIS related to sum(Fibonacci(kn))(although not defined as such) are
k=1... A000071 (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)= lucas(k)*a(n-1) +(-1)^(k+1)*a(n-2)+Fibonacci(k), n>1
The Koshy reference gives the closed form
sum(Fibonacci(k*j),k=0..n) = (Fibonacci(nk+k)-(-1)^k*Fibonacci(nk)-Fibonacci(k))/( L(k)-(-1)^k-1 ), where L(n)= A000032(n)
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REFERENCES
| Th. Koshy; Fibonacci and Lucas numbers with applications, Wiley,2001, p. 86.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (12,-10,-1).
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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-2), n>2.
a(n)= 1/11*(Fibonacci(5n+5)+ Fibonacci(5n)-5).
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MAPLE
| with(combinat):fs5:=n-> sum(fibonacci(5*k), k=0..n):
seq(fs5(n), n=0..18)
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CROSSREFS
| Sequence in context: A126275 A059602 A099672 * A093885 A192948 A156125
Adjacent sequences: A138131 A138132 A138133 * A138135 A138136 A138137
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KEYWORD
| nonn
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AUTHOR
| Gary Detlefs (gdetlefs(AT)aol.com), Dec 07 2010
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