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A005831
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a(n+1)=a(n)(a(n-1)+1).
(Formerly M1264)
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2
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0, 1, 1, 2, 4, 12, 60, 780, 47580, 37159980, 1768109008380, 65702897157329640780, 116169884340604934905464739377180, 7632697963609645128663145969343357330533515068777580
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| A discrete analogue of the derivative of t(x) = tetration base e, since t'(x) = t(x) * t(x-1) * t(x-2) * ... y = y * exp(y) * exp(exp(y)) * ... * t(x) This sequence satisfies almost the same equation but the derivative is replaced by a difference, comparable to the relations between differential equations and their associated difference equations. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008
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REFERENCES
| M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| a(0) = a(1) = 1, a(2) = 2; a(n) = a(n-1)*a(n-2)*a(n-3)*... + a(n-1). - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008
The sequence grows like a doubly exponential function, similar to Sylvester's sequence. In fact we have the asymptotic form : a(n) ~ e ^ (Phi ^ n) where e and Phi are the best possible constants. - Raes Tom (tommy1729(AT)hotmail.com), Aug 06 2008
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EXAMPLE
| a(5) = 12 since 12 = 1*2*4 + 4
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MATHEMATICA
| a=0; b=1; lst={a, b}; Do[c=a*b+b; AppendTo[lst, c]; a=b; b=c, {n, 3*3!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Sep 13 2009]
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PROG
| (Haskell)
a005831 n = a005831_list !! n
a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)
-- Reinhard Zumkeller, Mar 19 2011
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CROSSREFS
| Cf. A007807.
Cf. A000058, A001622, A001113, A102575, A096436, A111129.
Sequence in context: A118456 A013202 A004400 * A136512 A137160 A013207
Adjacent sequences: A005828 A005829 A005830 * A005832 A005833 A005834
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2008 at the suggestion of R. J. Mathar
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