A059727 - OEIS

%I #51 Jun 30 2026 19:30:12

%S 0,2,2,6,12,30,72,182,462,1190,3080,8010,20880,54522,142506,372710,

%T 975156,2552006,6679640,17484942,45771990,119825862,313697232,

%U 821252306,2150037792,5628825650,14736381842,38580227142,101004149532,264431978670,692291393640,1812441566630

%N a(n) = Fibonacci(n)*(Fibonacci(n) + 1).

%D L. Euler, Observationes analyticae, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 15, p. 54.

%H 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 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-5,-1,1).

%F G.f.: 2*x*(1 - 2*x - x^2 + x^3)/((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).

%F a(n) = Fibonacci(n) + (1/5)*(Lucas(2*n) - 2*(-1)^n).

%t #(#+1)&/@Fibonacci[Range[0,40]] (* _Harvey P. Dale_, May 29 2025 *)

%t (* Alternative: *)

%t LinearRecurrence[{3,1,-5,-1,1},{0,2,2,6,12},40] (* _Harvey P. Dale_, May 29 2025 *)

%o (PARI) a(n)=2*binomial(fibonacci(n)+1,2)

%o (PARI) a(n) = { my(f=fibonacci(n)); f*(f + 1) } \\ _Harry J. Smith_, Jun 29 2009

%o (Magma) [ Fibonacci(n)*(Fibonacci(n)+1): n in [0..100]]; // _Vincenzo Librandi_, Apr 15 2011

%o (Haskell)

%o a059727 n = a059727_list !! n

%o a059727_list = zipWith (*) a000045_list $ map (+ 1) a000045_list

%o -- _Reinhard Zumkeller_, Dec 17 2011

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 09 2001