|
%I #40 Sep 08 2022 08:45:25
%S 1,5,14,39,103,272,713,1869,4894,12815,33551,87840,229969,602069,
%T 1576238,4126647,10803703,28284464,74049689,193864605,507544126,
%U 1328767775,3478759199,9107509824,23843770273,62423800997,163427632718
%N Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).
%H Vincenzo Librandi, <a href="/A119996/b119996.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).
%F 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
%F G.f.: ((x-2)*x-1)/(x^4 - 3*x^3 + 3*x - 1). - _Harvey P. Dale_, Aug 22 2011
%F a(n) = Fibonacci(n+1)*Fibonacci(n+2) - 1. - _Gary Detlefs_, Mar 31 2012
%F a(n) = Sum_{k=1..n} Fibonacci(k+1)^2. Can be proved by induction from Gary Detlefs formula. - _Joel Courtheyn_, Mar 15 2021
%p with(combinat): seq(fibonacci(n+1)*fibonacci(n+2)-1, n=1..30); # _Zerinvary Lajos_, Jan 31 2008
%t Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]),{k,n}],{n,30}]]
%t LinearRecurrence[{3,0,-3,1},{1,5,14,39},30] (* _Harvey P. Dale_, Aug 22 2011 *)
%o (Magma) [Fibonacci(n+1)* Fibonacci(n+2)-1: n in [1..30]]; // _Vincenzo Librandi_, Aug 14 2012
%o (PARI) vector(30, n, f=fibonacci; f(n+1)*f(n+2)-1) \\ _G. C. Greubel_, Jul 23 2019
%o (Sage) f=fibonacci; [f(n+1)*f(n+2)-1 for n in (1..30)] # _G. C. Greubel_, Jul 23 2019
%o (GAP) F:=Fibonacci;; List([1..30], n-> F(n+1)*F(n+2)-1); # _G. C. Greubel_, Jul 23 2019
%Y Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.
%K frac,nonn,easy
%O 1,2
%A _Alexander Adamchuk_, Aug 03 2006
|
