A065563 Product of three consecutive Fibonacci numbers.

2, 6, 30, 120, 520, 2184, 9282, 39270, 166430, 704880, 2986128, 12649104, 53583010, 226980390, 961505790, 4073001576, 17253515288, 73087057560, 309601753890, 1311494059590, 5555578014142, 23533806080736, 99690802394400, 422297015565600, 1788878864806850, 7577812474550214

Offset: 1

Author

Len Smiley, Nov 30 2001

Keywords

• nonn
• easy

References

• Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
• Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
• T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 89, No. 32, with a minus sign.

Links

• Harry J. Smith, Table of n, a(n) for n=1..200
• V. E. Hoggatt and D. A. Lind, The Heights of Fibonacci Polynomials and an Associated Function, Fibonacci Quarterly, Vol. 5, No. 2 (April, 1967), pp. 141-152.
• Joseph S. Ozbolt, A New Sequence Derived From a Combination of Cubes with Volume Fn^3, Fibonacci Quarterly, Vol. 50, No. 1 (2012), pp. 19-26.
• Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Crossrefs

Cf. A000045, A001655, A079586, A324007.

Programs

Magma

[&*[Fibonacci(n+k): k in [0..2] ]: n in [1..30]]; // Vincenzo Librandi, Apr 09 2020

Maple

with (combinat):a:=n->fibonacci(n)*fibonacci(n+1)*fibonacci(n+2): seq(a(n), n=1..22); # Zerinvary Lajos, Oct 07 2007

Mathematica

Times@@@Partition[Fibonacci[Range[30]],3,1] (* Harvey P. Dale, Aug 18 2011 *)

PARI

a(n) = { fibonacci(n)*fibonacci(n + 1)*fibonacci(n + 2) } \\ Harry J. Smith, Oct 22 2009

Formula

a(n) = A000045(n)*A000045(n+1)*A000045(n+2).

G.f.: 2/(1 - 3*x - 6*x^2 + 3*x^3 + x^4).

a(n) = 2*A001655(n).

a(n) = Fibonacci(n+1)^3-(-1)^n*Fibonacci(n+1). - Gary Detlefs, Feb 02 2011

This corrects a sign mistake in the Koshy reference. - Wolfdieter Lang, Aug 07 2012

a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4).

O.g.f.: 2*x/((1 + x - x^2)*(1 - 4*x - x^2)) (compare with A001655). - Wolfdieter Lang, Aug 06 2012

Sum_{n>=1} (-1)^(n+1)/a(n) = A079586 - 3. - Amiram Eldar, Oct 04 2020

Sum_{n>=1} 1/a(n) = A324007. - Amiram Eldar, Feb 09 2023

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Oct 22 2009