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A187780
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Sum_{k=0..n} Lucas(k)^(n-k).
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2
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1, 3, 6, 13, 31, 84, 271, 1111, 6096, 44965, 434321, 5388944, 85434621, 1727597731, 44466614106, 1455616862597, 60619117448531, 3211943842710212, 216483614502128251, 18558646821817827015, 2023790814160269113876, 280732940929438329958733, 49535201863823517417076181
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) ~ c * ((1+sqrt(5))/2)^(n^2/4), where c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-k^2) = 2.555093469444518777230568... if n is even and c = Sum_{k=-Infinity..Infinity} ((1+sqrt(5))/2)^(-(k+1/2)^2) = 2.555093456793304790966994... if n is odd
G.f.: A(x) = Sum_{n>=0} x^n/(1 - Lucas(n)*x).
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MATHEMATICA
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Table[Sum[LucasL[k]^(n-k), {k, 0, n}], {n, 0, 20}]
(* constants: *)
ceven = N[Sum[((1+Sqrt[5])/2)^(-k^2), {k, -Infinity, +Infinity}], 50]
codd = N[Sum[((1+Sqrt[5])/2)^(-(k+1/2)^2), {k, -Infinity, +Infinity}], 50]
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PROG
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(PARI) Lucas(n)=fibonacci(n-1)+fibonacci(n+1)
a(n)=sum(k=0, n, Lucas(k)^(n-k))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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