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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 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 a(n) = Sum_{k=1..n} Fibonacci(k+1)^2. Can be proved by induction from Gary Detlefs formula. - Joel Courtheyn, Mar 15 2021 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 December 1 18:36 EST 2023. Contains 367500 sequences. (Running on oeis4.)