cp's OEIS Frontend

A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).

%I A305412 #24 Sep 08 2022 08:46:21
%S A305412 1,3,5,11,23,53,125,307,769,1959,5039,13049,33929,88451,230957,603667,
%T A305412 1578823,4130829,10810469,28295411,74067401,193893263,507590495,
%U A305412 1328842801,3478880593,9107706243,23844088085,62424315227,163428464759,427860443429,1120151837069
%N A305412 a(n) = F(n)*F(n+1) + F(n+2), where F = A000045 (Fibonacci numbers).
%H A305412 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).
%F A305412 G.f.: (1 - 5*x^2 - 2*x^3 + x^4)/((x + 1)*(1 - 3*x + x^2)*(1 - x - x^2)).
%F A305412 a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5).
%F A305412 5*a(n) = (-1)^(n+1) +5*F(n+2) + A002878(n). - _R. J. Mathar_, Nov 14 2019
%t A305412 Table[Fibonacci[n] Fibonacci[n+1] + Fibonacci[n+2], {n, 0, 30}]
%o A305412 (Magma) [Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2): n in [0..30]];
%o A305412 (GAP) List([0..35], n -> Fibonacci(n)*Fibonacci(n+1)+Fibonacci(n+2)); # _Muniru A Asiru_, Jun 06 2018
%Y A305412 Cf. A001654, A269803.
%Y A305412 Cf. A059769: F(n)*F(n+1) - F(n+2), with offset 3.
%Y A305412 Equals A000045 + A286983.
%Y A305412 First differences are listed in A059727 (after 0).
%K A305412 nonn,easy
%O A305412 0,2
%A A305412 _Vincenzo Librandi_, Jun 05 2018