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

Author

N. J. A. Sloane, Jun 11 2012

Keywords

• nonn

Comments

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.

References

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

Links

• 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!)

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

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

Formula

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

Extensions

Definition recovered by Johan de Ruiter, Jul 23 2012