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
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar
STATUS
approved