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

Offset

1,1

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).

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

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 *)

Prog

(PARI) { for (n=1, 200, a=fibonacci(n)*fibonacci(n + 1)*fibonacci(n + 2); write("b065563.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 22 2009
(Magma) [&*[Fibonacci(n+k): k in [0..2] ]: n in [1..30]]; // Vincenzo Librandi, Apr 09 2020

Crossrefs

Cf. A000045, A001655, A079586, A324007.

Sequence in context: A071760 A304579 A036752 * A035105 A073969 A295863

Adjacent sequences: A065560 A065561 A065562 * A065564 A065565 A065566

Keyword

nonn,easy

Author

Len Smiley, Nov 30 2001

Extensions

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

Status

approved