A000289 - OEIS

%I M3316 N1333 #53 Apr 07 2021 02:49:57

%S 1,4,7,31,871,756031,571580604871,326704387862983487112031,

%T 106735757048926752040856495274871386126283608871,

%U 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068031

%N A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).

%C An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004

%C This is the special case k=3 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215. - _Seppo Mustonen_, Sep 04 2005

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Cerkan, <a href="/A000289/b000289.txt">Table of n, a(n) for n = 0..12</a>

%H A. V. Aho and N. J. A. Sloane, <a href="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, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2311857">On certain nonlinear recurring sequences</a>, Amer. Math. Monthly 70 (1963), 403-405.

%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012 - From N. J. A. Sloane, Jun 13 2012

%H S. Mustonen, <a href="http://www.survo.fi/papers/resseq.pdf">On integer sequences with mutual k-residues</a>

%H Seppo Mustonen, <a href="/A000215/a000215.pdf">On integer sequences with mutual k-residues</a> [Local copy]

%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>

%F a(n) = A005267(n) + 2 (for n>0).

%F a(n) = ceiling(c^(2^n)) + 1 where c = A077141. - _Benoit Cloitre_, Nov 29 2002

%F For n>0, a(n) = 3 + Product_{i=0..n-1} a(i). - _Vladimir Shevelev_, Dec 08 2010

%t Join[{1}, RecurrenceTable[{a[n] == a[n-1]^2 - 3*a[n-1] + 3, a[1] == 4}, a, {n, 1, 9}]] (* _Jean-François Alcover_, Feb 06 2016 *)

%o (PARI) a(n)=if(n<2,max(0,1+3*n),a(n-1)^2-3*a(n-1)+3)

%Y Cf. A000058.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_