A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).
1, 1, 1, 2, 6, 30, 240, 3120, 65520, 2227680, 122522400, 10904493600, 1570247078400, 365867569267200, 137932073613734400, 84138564904377984000, 83044763560621070208000, 132622487406311849122176000, 342696507457909818131702784000
Offset: 0
Examples
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 + ...
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.
- Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
- Tipaluck Krityakierne and Thotsaporn Aek Thanatipanonda, Ansatz in a Nutshell: A Comprehensive Step-by-Step Guide to Polynomial, C-finite, Holonomic, and C^2-finite Sequences, in Applied Mathematical Analysis and Computations (SGMC 2021) Springer Proc. Math. Stat., Vol. 471. Springer, Cham, 255-297. See p. 287.
- 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.
- Aidan Sudbury, Arthur Sun, David Treeby, and Edward Wang, Pick-up Sticks and the Fibonacci Factorial, arXiv:2504.19911 [math.PR], 2025.
- Thotsaporn Aek Thanatipanonda and Yi Zhang, Sequences: Polynomial, C-finite, Holonomic, ..., arXiv:2004.01370 [math.CO], 2020. See pp. 5-6.
- Eric Weisstein's World of Mathematics, Fibonorial
- Index to divisibility sequences.
Crossrefs
Programs
-
Haskell
a003266 n = a003266_list !! (n-1) a003266_list = scanl1 (*) $ tail a000045_list -- Reinhard Zumkeller, Sep 03 2013
-
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 *)
-
PARI
a(n)=prod(i=1,n,fibonacci(i)) \\ Charles R Greathouse IV, Jan 13 2012
-
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
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
a(n) = (2/sqrt(5))^n * Product_{j=1..n} i^j*sinh(c*j), where c = arccsch(2) - i*Pi/2. - Peter Luschny, Jul 07 2025
Extensions
a(0)=1 prepended by Alois P. Heinz, Oct 12 2016
Comments