A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n). (Formerly M1692)

1, 1, 1, 2, 6, 30, 240, 3120, 65520, 2227680, 122522400, 10904493600, 1570247078400, 365867569267200, 137932073613734400, 84138564904377984000, 83044763560621070208000, 132622487406311849122176000, 342696507457909818131702784000

Offset

0,4

Comments

Equals right border of unsigned triangle A158472. - Gary W. Adamson, Mar 20 2009

Three closely related sequences are A194157 (product of first n nonzero F(2*n)), A194158 (product of first n nonzero F(2*n-1)) and A123029 (a(2*n) = A194157(n) and a(2*n-1) = A194158(n)). - Johannes W. Meijer, Aug 21 2011

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, second edition, Addison Wesley, p 597

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..99 (terms n = 1..50 from T. D. Noe)

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 74.

Spencer J. Franks, Pamela E. Harris, Kimberly Harry, Jan Kretschmann, and Megan Vance, Counting Parking Sequences and Parking Assortments Through Permutations, arXiv:2301.10830 [math.CO], 2023.

Mathematica Stack Exchange, Product of Fibonacci numbers using For/Do/While loops.

Yuri V. Matiyasevich and Richard K. Guy, A new formula for pi, Amer. Math. Monthly 93 (1986), no. 8, 631-635. Math. Rev. 2000i:11199.

Thotsaporn Aek Thanatipanonda and Yi Zhang, Sequences: Polynomial, C-finite, Holonomic, ..., arXiv:2004.01370 [math.CO], 2020.

Eric Weisstein's World of Mathematics, Fibonorial

Index to divisibility sequences.

Formula

a(n) is asymptotic to C*phi^(n*(n+1)/2)/sqrt(5)^n where phi = (1 + sqrt(5))/2 is the golden ratio and the decimal expansion of C is given in A062073. - Benoit Cloitre, Jan 11 2003

a(n+3) = a(n+1)*a(n+2)/a(n) + a(n+2)^2/a(n+1). - Robert Israel, May 19 2014

a(0) = 1 by convention since empty products equal 1. - Michael Somos, Oct 06 2014

0 = a(n)*(+a(n+1)*a(n+3) - a(n+2)^2) + a(n+2)*(-a(n+1)^2) for all n >= 0. - Michael Somos, Oct 06 2014

Sum_{n>=1} 1/a(n) = A101689. - Amiram Eldar, Oct 27 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = A135598. - Amiram Eldar, Apr 12 2021

Example

a(5) = 30 because the first 5 Fibonacci numbers are 1, 1, 2, 3, 5 and 1 * 1 * 2 * 3 * 5 = 30.
a(6) = 240 because 8 is the sixth Fibonacci number and a(5) * 8 = 240.
a(7) = 3120 because 13 is the seventh Fibonacci number and a(6) * 13 = 3120.
G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 30*x^5 + 240*x^6 + 3120*x^7 + ...

Maple

with(combinat): A003266 := n-> mul(fibonacci(i), i=1..n): seq(A003266(n), n=0..20);

Mathematica

Rest[FoldList[Times, 1, Fibonacci[Range[20]]]] (* Harvey P. Dale, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, Fibonorial[n]]; (* Michael Somos, Oct 23 2017 *)
Table[Round[GoldenRatio^(n(n-1)/2) QFactorial[n, GoldenRatio-2]], {n, 20}] (* Vladimir Reshetnikov, Sep 14 2016 *)

Prog

(PARI) a(n)=prod(i=1, n, fibonacci(i)) \\ Charles R Greathouse IV, Jan 13 2012
(Haskell)
a003266 n = a003266_list !! (n-1)
a003266_list = scanl1 (*) $ tail a000045_list
-- Reinhard Zumkeller, Sep 03 2013
(Python)
from itertools import islice
def A003266_gen(): # generator of terms
    a, b, c = 1, 1, 1
    while True:
        yield c
        c *= a
        a, b = b, a+b
A003266_list = list(islice(A003266_gen(), 20)) # Chai Wah Wu, Jan 11 2023

Crossrefs

Cf. A000045, A101689, A135598, A158472.

Cf. A123741 (for Fibonacci second version), A002110 (for primes), A070825 (for Lucas), A003046 (for Catalan), A126772 (for Padovan), A069777 (q-factorial numbers for sums of powers). - Johannes W. Meijer, Aug 21 2011]

Cf. A176343, A238243, A238244.

Sequence in context: A027882 A306782 A106209 * A303169 A369198 A097385

Adjacent sequences: A003263 A003264 A003265 * A003267 A003268 A003269

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 12 2016

Status

approved