A065563 - OEIS

%I #69 Jan 05 2025 19:51:36

%S 2,6,30,120,520,2184,9282,39270,166430,704880,2986128,12649104,

%T 53583010,226980390,961505790,4073001576,17253515288,73087057560,

%U 309601753890,1311494059590,5555578014142,23533806080736,99690802394400,422297015565600,1788878864806850,7577812474550214

%N Product of three consecutive Fibonacci numbers.

%D Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.

%D Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.

%D T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001, p. 89, No. 32, with a minus sign.

%H Harry J. Smith, <a href="/A065563/b065563.txt">Table of n, a(n) for n=1..200</a>

%H V. E. Hoggatt and D. A. Lind, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-2/hoggatt.pdf">The Heights of Fibonacci Polynomials and an Associated Function</a>, Fibonacci Quarterly, Vol. 5, No. 2 (April, 1967), pp. 141-152.

%H Joseph S. Ozbolt, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/50-1/Ozbolt.pdf">A New Sequence Derived From a Combination of Cubes with Volume Fn^3</a>, Fibonacci Quarterly, Vol. 50, No. 1 (2012), pp. 19-26.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-3,-1).

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

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

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

%F a(n) = Fibonacci(n+1)^3-(-1)^n*Fibonacci(n+1). - _Gary Detlefs_, Feb 02 2011

%F This corrects a sign mistake in the Koshy reference. - _Wolfdieter Lang_, Aug 07 2012

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

%F O.g.f.: 2*x/((1 + x - x^2)*(1 - 4*x - x^2)) (compare with A001655). - _Wolfdieter Lang_, Aug 06 2012

%F Sum_{n>=1} (-1)^(n+1)/a(n) = A079586 - 3. - _Amiram Eldar_, Oct 04 2020

%F Sum_{n>=1} 1/a(n) = A324007. - _Amiram Eldar_, Feb 09 2023

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

%t Times@@@Partition[Fibonacci[Range[30]],3,1] (* _Harvey P. Dale_, Aug 18 2011 *)

%o (PARI) a(n) = { fibonacci(n)*fibonacci(n + 1)*fibonacci(n + 2) } \\ _Harry J. Smith_, Oct 22 2009

%o (Magma) [&*[Fibonacci(n+k): k in [0..2] ]: n in [1..30]]; // _Vincenzo Librandi_, Apr 09 2020

%Y Cf. A000045, A001655, A079586, A324007.

%K nonn,easy

%O 1,1

%A _Len Smiley_, Nov 30 2001

%E Offset changed from 0 to 1 by _Harry J. Smith_, Oct 22 2009