A214636 -id:A214636 - cp's OEIS frontend

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

1, 3, 7, 31, 703, 459007, 210066847231, 44127887746116242376703, 1947270476915296449559747573381594836628779007

Offset: 1

N. J. A. Sloane, Jun 11 2012

nonn

This sequence was going to be included in the Aho-Sloane paper, but was omitted from the published version.

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437. [Includes many similar sequences, although not this one.]
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)

Cf. A076949, A214635, A214636, A003095, A001699.

A213437 := proc(n)
        if n = 1 then 1;
        else procname(n-1)+(1+procname(n-1))*mul(procname(j),j=1..n-2);
        end if;
end proc: # R. J. Mathar, Jul 23 2012

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 *)

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

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
a(2+3k) = 9007 (mod 10^4) for all k>0. - M. F. Hasler, Jul 23 2012
a(n) ~ c^(2^n), where c = A076949 = 1.2259024435287485386279474959130085213212293209696612823177009... . - Vaclav Kotesovec, May 06 2015
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

Definition recovered by Johan de Ruiter, Jul 23 2012

A214635 Period of A213437 mod n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 4, 1, 3, 1, 6, 3, 1, 3, 1, 1, 1, 1, 3, 4, 3, 1, 1, 3, 1, 1, 1, 6, 3, 3, 1, 1, 4, 3, 7, 1, 1, 1, 3, 1, 4, 1, 6, 3, 6, 4, 1, 3, 3, 1, 1, 1, 3, 3, 10, 1, 3, 1, 12, 1, 1, 6, 1, 3, 11, 3, 6, 1, 3, 1, 1, 4, 4, 3, 9, 7, 5, 1, 6, 1, 1, 1, 14, 3, 4, 1, 1, 4, 3, 1, 3, 6, 3

Offset: 1

David Applegate, M. F. Hasler and N. J. A. Sloane, Jul 23 2012

nonn

Cf. A213437, A214636.

A214635(n,N=99)={my(a=[Mod(1,n)]); for(n=1,N-1,a=concat(a,a[n]+(a[n]+1)*prod(k=1,n-1,a[k])));for(p=1,N\3,forstep(m=N,p+1,-1,a[m]==a[m-p]&next;3*m>N&next(2);return(p));return(p))} /* the 2nd optional parameter must be taken large enough, at least 3 times the period length and starting position. The script returns zero if the period is not found (probably due to these constraints). */

Empirically:
A214635(2^n) = 1,  A214635(5^n) = A214635(10^n) = 3, for all n>0.
A214635(3^n) = A214635(6^n) = (1, 3, 3, 9, 27, 81, ...) = 3^(n-2) for n>2.
A214635(15^n) = (3,3,3,9,27,81,...) = A214635(3^n) for n>1.
A214635(7^n) = (1,6,42,294,...) = 6*7^(n-2) for n>1.
A214635(11^n) = (1,20,220,2420,...) = 20*11^(n-2) for n>1. - M. F. Hasler, Jul 24 2012

cp's OEIS Frontend

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

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