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A003266
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Product of first n nonzero Fibonacci numbers F(1), ..., F(n).
(Formerly M1692)
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83
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1, 1, 1, 2, 6, 30, 240, 3120, 65520, 2227680, 122522400, 10904493600, 1570247078400, 365867569267200, 137932073613734400, 84138564904377984000, 83044763560621070208000, 132622487406311849122176000, 342696507457909818131702784000
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OFFSET
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0,4
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COMMENTS
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a(n+1)^2 is the number of ways to tile this pyramid of height n with squares and dominoes, where vertical dominoes can only appear (if at all) in the central column. Here is a pyramid of height n=4,
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and here is one of the a(5)^2 = 900 possible such tilings with our given restrictions:
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REFERENCES
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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).
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LINKS
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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.
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FORMULA
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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
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EXAMPLE
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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 + ...
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MAPLE
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with(combinat): A003266 := n-> mul(fibonacci(i), i=1..n): seq(A003266(n), n=0..20);
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a003266 n = a003266_list !! (n-1)
a003266_list = scanl1 (*) $ tail a000045_list
(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
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CROSSREFS
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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