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A119996
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Numerator of Sum[ 1 / ( Fibonacci[k] * Fibonacci[k+2] ),{k,1,n}].
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5
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1, 5, 14, 39, 103, 272, 713, 1869, 4894, 12815, 33551, 87840, 229969, 602069, 1576238, 4126647, 10803703, 28284464, 74049689, 193864605, 507544126, 1328767775, 3478759199, 9107509824, 23843770273, 62423800997, 163427632718
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Prime p divides a(p-1) for p = {11,19,29,31,41,59,61,71,79,89,101,109,...} = A045468[n] Primes congruent to {1, 4} mod 5 or (except for the first term =5) A064739[n] Primes p such that Fibonacci(p)-1 is divisible by p. Prime p divides a((p-1)/2) for p = {5,7,29,41,61,89,101,109,149,181,229,241,269,281,349,389,401,409,421,449,461,509,521,541,...} Primes congruent to 1, 5, 9 (mod 20) or only Prime Norms of prime elements of Z[sqrt(-5)] A091729[n] (excluding squares). Prime p divides a((p-1)/3) for p = {13,23,41,139,151,199,331,541,...}.
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FORMULA
| a(n) = Numerator[Sum[1/(Fibonacci[k]*Fibonacci[k+2]),{k,1,n}]].
a(0)=1, a(1)=5, a(2)=14, a(3)=39, a(n)=3*a(n-1)-3*a(n-3)+a(n-4) [From Harvey P. Dale, Aug 22 2011]
G.f.: ((x-2)*x-1)/(x^4-3*x^3+3*x-1) [From Harvey P. Dale, Aug 22 2011]
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MAPLE
| with(combinat): seq(fibonacci(n)*fibonacci(n+1)-1, n=2..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008
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MATHEMATICA
| Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]), {k, 1, n}], {n, 1, 50}]]
LinearRecurrence[{3, 0, -3, 1}, {1, 5, 14, 39}, 50] (* From Harvey P. Dale, Aug 22 2011 *)
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CROSSREFS
| Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.
Sequence in context: A183898 A111715 A024525 * A027089 A184437 A023871
Adjacent sequences: A119993 A119994 A119995 * A119997 A119998 A119999
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KEYWORD
| frac,nonn
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AUTHOR
| Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 03 2006
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