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A005831 a(n+1) = a(n) * (a(n-1) + 1).
(Formerly M1264)
3
0, 1, 1, 2, 4, 12, 60, 780, 47580, 37159980, 1768109008380, 65702897157329640780, 116169884340604934905464739377180, 7632697963609645128663145969343357330533515068777580, 886689639639303288926299195509965193299034793881606681727875910370940270908216401980 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A discrete analog 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

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n = 0..19

M. E. Mays, Iterating the division algorithm, Fib. Quart., 25 (1987), 204-213.

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

EXAMPLE

a(5) = 12 since 12 = 1*2*4 + 4.

MATHEMATICA

a=0; b=1; lst={a, b}; Do[c=a*b+b; AppendTo[lst, c]; a=b; b=c, {n, 18}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1](a[n-2]+1)}, a, {n, 15}] (* Harvey P. Dale, Aug 17 2013 *)

PROG

(Haskell)

a005831 n = a005831_list !! n

a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)

-- Reinhard Zumkeller, Mar 19 2011

CROSSREFS

Cf. A007807.

Cf. A000058, A001622, A001113, A102575, A096436, A111129.

Sequence in context: A013202 A253832 A004400 * A136512 A137160 A217716

Adjacent sequences:  A005828 A005829 A005830 * A005832 A005833 A005834

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jeffrey Shallit

EXTENSIONS

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

STATUS

approved

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Last modified July 6 23:56 EDT 2020. Contains 335484 sequences. (Running on oeis4.)