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

0, 2, 2, 6, 12, 30, 72, 182, 462, 1190, 3080, 8010, 20880, 54522, 142506, 372710, 975156, 2552006, 6679640, 17484942, 45771990, 119825862, 313697232, 821252306, 2150037792, 5628825650, 14736381842, 38580227142, 101004149532, 264431978670, 692291393640, 1812441566630

Offset: 0

Author

N. J. A. Sloane, Feb 09 2001

Keywords

• nonn
• easy

References

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

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..300 [replacing table for n = 0..200 by Harry J. Smith]
• G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
• Eric Weisstein's World of Mathematics, Trinomial Coefficient.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Programs

Haskell

a059727 n = a059727_list !! n
a059727_list = zipWith (*) a000045_list $ map (+ 1) a000045_list
-- Reinhard Zumkeller, Dec 17 2011

Magma

[ Fibonacci(n)*(Fibonacci(n)+1): n in [0..100]]; // Vincenzo Librandi, Apr 15 2011

Mathematica

#(#+1)&/@Fibonacci[Range[0,40]] (* or *) LinearRecurrence[{3,1,-5,-1,1},{0,2,2,6,12},40] (* Harvey P. Dale, May 29 2025 *)

PARI

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

PARI

a(n) = { my(f=fibonacci(n)); f*(f + 1) } \\ Harry J. Smith, Jun 29 2009

Formula

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

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