cp's OEIS Frontend

A059727 a(n) = Fibonacci(n)*(Fibonacci(n) + 1).

%I A059727 #49 May 29 2025 12:06:36
%S A059727 0,2,2,6,12,30,72,182,462,1190,3080,8010,20880,54522,142506,372710,
%T A059727 975156,2552006,6679640,17484942,45771990,119825862,313697232,
%U A059727 821252306,2150037792,5628825650,14736381842,38580227142,101004149532,264431978670,692291393640,1812441566630
%N A059727 a(n) = Fibonacci(n)*(Fibonacci(n) + 1).
%D A059727 L. Euler, Observationes analyticae, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 54.
%H A059727 Nathaniel Johnston, <a href="/A059727/b059727.txt">Table of n, a(n) for n = 0..300</a> [replacing table for n = 0..200 by Harry J. Smith]
%H A059727 G. E. Andrews, <a href="http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html">Three aspects of partitions</a>, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
%H A059727 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H A059727 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).
%F A059727 G.f.: 2*x*(1 - 2*x - x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).
%F A059727 a(n) = Fibonacci(n) + (1/5)*(Lucas(2*n) - 2*(-1)^n).
%t A059727 #(#+1)&/@Fibonacci[Range[0,40]] (* or *) LinearRecurrence[{3,1,-5,-1,1},{0,2,2,6,12},40] (* _Harvey P. Dale_, May 29 2025 *)
%o A059727 (PARI) a(n)=2*binomial(fibonacci(n)+1,2)
%o A059727 (PARI) a(n) = { my(f=fibonacci(n)); f*(f + 1) } \\ _Harry J. Smith_, Jun 29 2009
%o A059727 (Magma) [ Fibonacci(n)*(Fibonacci(n)+1): n in [0..100]]; // _Vincenzo Librandi_, Apr 15 2011
%o A059727 (Haskell)
%o A059727 a059727 n = a059727_list !! n
%o A059727 a059727_list = zipWith (*) a000045_list $ map (+ 1) a000045_list
%o A059727 -- _Reinhard Zumkeller_, Dec 17 2011
%K A059727 nonn,easy
%O A059727 0,2
%A A059727 _N. J. A. Sloane_, Feb 09 2001