login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A003266 Product of first n nonzero Fibonacci numbers F(1), ..., F(n).
(Formerly M1692)
70

%I M1692

%S 1,1,1,2,6,30,240,3120,65520,2227680,122522400,10904493600,

%T 1570247078400,365867569267200,137932073613734400,

%U 84138564904377984000,83044763560621070208000,132622487406311849122176000,342696507457909818131702784000

%N Product of first n nonzero Fibonacci numbers F(1), ..., F(n).

%C Equals right border of unsigned triangle A158472. - _Gary W. Adamson_, Mar 20 2009

%C 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

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

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

%H T. D. Noe and Alois P. Heinz, <a href="/A003266/b003266.txt">Table of n, a(n) for n = 0..99</a> (terms n = 1..50 from T. D. Noe)

%H Alfred Brousseau, <a href="http://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972, p. 74.

%H Yuri V. Matiyasevich and Richard K. Guy, <a href="http://www.jstor.org/stable/2322322">A new formula for pi</a>, Amer. Math. Monthly 93 (1986), no. 8, 631-635. Math. Rev. 2000i:11199.

%H Stackexchange, <a href="http://mathematica.stackexchange.com/questions/19637/">Product of Fibonacci numbers using For/Do/While loops</a>.

%H Thotsaporn Aek Thanatipanonda and Yi Zhang, <a href="https://arxiv.org/abs/2004.01370">Sequences: Polynomial, C-finite, Holonomic, ...</a>, arXiv:2004.01370 [math.CO], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fibonorial.html">Fibonorial</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.

%F 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

%F a(n+3) = a(n+1)*a(n+2)/a(n) + a(n+2)^2/a(n+1). - _Robert Israel_, May 19 2014

%F a(0) = 1 by convention since empty products equal 1. - _Michael Somos_, Oct 06 2014

%F 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

%F Sum_{n>=1} 1/a(n) = A101689. - _Amiram Eldar_, Oct 27 2020

%F Sum_{n>=1} (-1)^(n+1)/a(n) = A135598. - _Amiram Eldar_, Apr 12 2021

%e a(5) = 30 because the first 5 Fibonacci numbers are 1, 1, 2, 3, 5 and 1 * 1 * 2 * 3 * 5 = 30.

%e a(6) = 240 because 8 is the sixth Fibonacci number and a(5) * 8 = 240.

%e a(7) = 3120 because 13 is the seventh Fibonacci number and a(6) * 13 = 3120.

%e G.f. = 1 + x + x^2 + 2*x^3 + 6*x^4 + 30*x^5 + 240*x^6 + 3120*x^7 + ...

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

%t Rest[FoldList[Times,1,Fibonacci[Range[20]]]] (* _Harvey P. Dale_, Jul 11 2011 *)

%t a[ n_] := If[ n < 0, 0, Fibonorial[n]]; (* _Michael Somos_, Oct 23 2017 *)

%t Table[Round[GoldenRatio^(n(n-1)/2) QFactorial[n, GoldenRatio-2]], {n, 20}] (* _Vladimir Reshetnikov_, Sep 14 2016 *)

%o (PARI) a(n)=prod(i=1,n,fibonacci(i)) \\ _Charles R Greathouse IV_, Jan 13 2012

%o (Haskell)

%o a003266 n = a003266_list !! (n-1)

%o a003266_list = scanl1 (*) $ tail a000045_list

%o -- _Reinhard Zumkeller_, Sep 03 2013

%Y Cf. A000045, A101689, A135598, A158472.

%Y 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]

%Y Cf. A176343, A238243, A238244.

%K nonn,easy,nice,changed

%O 0,4

%A _N. J. A. Sloane_

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 12 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 15 17:25 EDT 2021. Contains 342977 sequences. (Running on oeis4.)