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A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)). 6
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; text; internal format)
OFFSET

1,2

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, ...}. [These comments are very hard to understand! - N. J. A. Sloane, Jan 19 2019]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3, 0, -3, 1).

FORMULA

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4); a(0)=1, a(1)=5, a(2)=14, a(3)=39. - Harvey P. Dale, Aug 22 2011

G.f.: ((x-2)*x-1)/(x^4 - 3*x^3 + 3*x - 1). - Harvey P. Dale, Aug 22 2011

a(n) = Fibonacci(n+1)*Fibonacci(n+2) - 1. - Gary Detlefs, Mar 31 2012

MAPLE

with(combinat): seq(fibonacci(n+1)*fibonacci(n+2)-1, n=1..30); # Zerinvary Lajos, Jan 31 2008

MATHEMATICA

Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]), {k, n}], {n, 30}]]

LinearRecurrence[{3, 0, -3, 1}, {1, 5, 14, 39}, 30] (* Harvey P. Dale, Aug 22 2011 *)

PROG

(MAGMA) [Fibonacci(n+1)* Fibonacci(n+2)-1: n in [1..30]]; // Vincenzo Librandi, Aug 14 2012

(PARI) vector(30, n, f=fibonacci; f(n+1)*f(n+2)-1) \\ G. C. Greubel, Jul 23 2019

(Sage) f=fibonacci; [f(n+1)*f(n+2)-1 for n in (1..30)] # G. C. Greubel, Jul 23 2019

(GAP) F:=Fibonacci;; List([1..30], n-> F(n+1)*F(n+2)-1); # G. C. Greubel, Jul 23 2019

CROSSREFS

Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.

Sequence in context: A111715 A024525 A209536 * A027089 A184437 A023871

Adjacent sequences:  A119993 A119994 A119995 * A119997 A119998 A119999

KEYWORD

frac,nonn,easy

AUTHOR

Alexander Adamchuk, Aug 03 2006

STATUS

approved

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Last modified October 18 12:18 EDT 2019. Contains 328160 sequences. (Running on oeis4.)