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A213437 Nonlinear recurrence: a(n) = a(n-1) + (a(n-1)+1)*Product_{j=1..n-2} a(j).

%I A213437 N1082 #50 Jan 09 2025 09:56:53
%S A213437 1,3,7,31,703,459007,210066847231,44127887746116242376703,
%T A213437 1947270476915296449559747573381594836628779007
%N A213437 Nonlinear recurrence: a(n) = a(n-1) + (a(n-1)+1)*Product_{j=1..n-2} a(j).
%C A213437 This sequence was going to be included in the Aho-Sloane paper, but was omitted from the published version.
%C A213437 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 A213437 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%H A213437 A. V. Aho and N. J. A. Sloane, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437. [Includes many similar sequences, although not this one.]
%H A213437 A. V. Aho and N. J. A. Sloane, <a href="/A000058/a000058.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
%F A213437 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 A213437 a(2+3k) = 9007 (mod 10^4) for all k>0. - _M. F. Hasler_, Jul 23 2012
%F A213437 a(n) ~ c^(2^n), where c = A076949 = 1.2259024435287485386279474959130085213212293209696612823177009... . - _Vaclav Kotesovec_, May 06 2015
%F A213437 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 A213437 := proc(n)
%p A213437         if n = 1 then 1;
%p A213437         else procname(n-1)+(1+procname(n-1))*mul(procname(j),j=1..n-2);
%p A213437         end if;
%p A213437 end proc: # _R. J. Mathar_, Jul 23 2012
%t A213437 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 A213437 (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 A213437 Cf. A076949, A214635, A214636, A003095, A001699.
%K A213437 nonn
%O A213437 1,2
%A A213437 _N. J. A. Sloane_, Jun 11 2012
%E A213437 Definition recovered by _Johan de Ruiter_, Jul 23 2012