%I N1082 #46 Mar 05 2020 17:26:58
%S 1,3,7,31,703,459007,210066847231,44127887746116242376703,
%T 1947270476915296449559747573381594836628779007
%N Nonlinear recurrence: a(n) = a(n-1) + (a(n-1)+1)*Product_{j=1..n-2} a(j).
%C This sequence was going to be included in the Aho-Sloane paper, but was omitted from the published version.
%C It appears that the sequence becomes periodic mod 10^k for any k, with period 3. The last digits are (1,3,7) repeated. Modulo 10^5 the sequence enters the cycle (56703, 79007, 23231) after the first 10 terms. - _M. F. Hasler_, Jul 23 2012. See also A214635, A214636.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A. V. Aho and N. J. A. Sloane, <a href="http://NeilSloane.com/doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., <a href="http://www.fq.math.ca/11-4.html">11 (1973)</a>, 429-437. [Includes many similar sequences, although not this one.]
%F a(n) = a(n-1)+(a(n-1)+1)*(a(n-1)-a(n-2))*a(n-2)/(a(n-2)+1). - _Johan de Ruiter_, Jul 23 2012
%F a(2+3k) = 9007 (mod 10^4) for all k>0. - _M. F. Hasler_, Jul 23 2012
%F a(n) ~ c^(2^n), where c = A076949 = 1.2259024435287485386279474959130085213212293209696612823177009... . - _Vaclav Kotesovec_, May 06 2015
%F a(n) = A001699(n)/A001699(n-1); a(n+1) - a(n) = A001699(n) + A001699(n-1); a(n) = A003095(n) + A003095(n-1). - _Peter Bala_, Feb 03 2017
%p A213437 := proc(n)
%p if n = 1 then 1;
%p else procname(n-1)+(1+procname(n-1))*mul(procname(j),j=1..n-2);
%p end if;
%p end proc: # _R. J. Mathar_, Jul 23 2012
%t RecurrenceTable[{a[n] == a[n-1]+(a[n-1]+1)*(a[n-1]-a[n-2])*a[n-2]/(a[n-2]+1),a[1]==1,a[2]==3},a,{n,1,10}] (* _Vaclav Kotesovec_, May 06 2015 *)
%o (PARI) a=[1];for(n=1,11,a=concat(a, a[n] + (a[n]+1) * prod(k=1,n-1, a[k] )));a \\ - _M. F. Hasler_, Jul 23 2012
%Y Cf. A076949, A214635, A214636, A003095, A001699.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Jun 11 2012
%E Definition recovered by _Johan de Ruiter_, Jul 23 2012