%I M1264 #29 Jan 05 2025 19:51:33

%S 0,1,1,2,4,12,60,780,47580,37159980,1768109008380,

%T 65702897157329640780,116169884340604934905464739377180,

%U 7632697963609645128663145969343357330533515068777580,886689639639303288926299195509965193299034793881606681727875910370940270908216401980

%N a(n+1) = a(n) * (a(n-1) + 1).

%C 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

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

%H T. D. Noe, <a href="/A005831/b005831.txt">Table of n, a(n) for n = 0..19</a>

%H M. E. Mays, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-3/mays.pdf">Iterating the division algorithm</a>, Fib. Quart., 25 (1987), 204-213.

%F 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

%F 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

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

%t 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 *)

%t 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 *)

%o (Haskell)

%o a005831 n = a005831_list !! n

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

%o -- _Reinhard Zumkeller_, Mar 19 2011

%Y Cf. A007807.

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

%K nonn,easy,nice

%O 0,4

%A _N. J. A. Sloane_, _Jeffrey Shallit_

%E Edited by _N. J. A. Sloane_, Aug 29 2008 at the suggestion of _R. J. Mathar_