This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A005831 M1264 #29 Jan 05 2025 19:51:33
%S A005831 0,1,1,2,4,12,60,780,47580,37159980,1768109008380,
%T A005831 65702897157329640780,116169884340604934905464739377180,
%U A005831 7632697963609645128663145969343357330533515068777580,886689639639303288926299195509965193299034793881606681727875910370940270908216401980
%N A005831 a(n+1) = a(n) * (a(n-1) + 1).
%C A005831 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 A005831 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005831 T. D. Noe, <a href="/A005831/b005831.txt">Table of n, a(n) for n = 0..19</a>
%H A005831 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 A005831 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 A005831 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 A005831 a(5) = 12 since 12 = 1*2*4 + 4.
%t A005831 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 A005831 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 A005831 (Haskell)
%o A005831 a005831 n = a005831_list !! n
%o A005831 a005831_list = 0:1:zipWith (*) (tail a005831_list) (map succ a005831_list)
%o A005831 -- _Reinhard Zumkeller_, Mar 19 2011
%Y A005831 Cf. A007807.
%Y A005831 Cf. A000058, A001622, A001113, A102575, A096436, A111129.
%K A005831 nonn,easy,nice
%O A005831 0,4
%A A005831 _N. J. A. Sloane_, _Jeffrey Shallit_
%E A005831 Edited by _N. J. A. Sloane_, Aug 29 2008 at the suggestion of _R. J. Mathar_