%I A002812 M1817 N0720
%S A002812 2,7,97,18817,708158977,1002978273411373057,
%T A002812 2011930833870518011412817828051050497,
%U A002812 8095731360557835890888779535060256832479295062749579257164654370487894017
%N A002812 a(n) = 2a(n-1)^2 - 1.
%C A002812 An infinite coprime sequence defined by recursion - Michael Somos, Mar 
               14 2004
%C A002812 2^p-1 is prime iff it divides a(p-2), since a(n) = A003010(n)/2, where 
               A003010 is the Lucas-Lehmer sequence used for Mersenne number primality 
               testing - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Mar 09 2007
%C A002812 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 
               2008: (Start)
%C A002812 Also numerators of the convergents to the square root of 3 using the
%C A002812 the following recursion for initial x = 1.
%C A002812 x1=x
%C A002812 x=3/x
%C A002812 x=(x+x1)/2
%C A002812 This recursion was derived by experimenting with polynomial recursions 
               of the
%C A002812 form x = -a(0)/(a(n-1)x^(n-1)+...+a(1)) in an effort to find a root for 
               the
%C A002812 polynomial a(n)x^n+a(n-1)x^(n-1)+...+a(0). The process was hit and miss 
               until
%C A002812 I introduced the averaging step described above. This method is equivelant 
               to
%C A002812 Newton's Method although derived somewhat differently. (End)
%D A002812 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002812 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002812 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 
               256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see 
               vol. 1, p. 399.
%D A002812 E. Lucas, Nouveaux theoremes d'arithmetique superieure, Comptes Rend., 
               83 (1876), 1286-1288.
%D A002812 M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, 
               Theoret. Comput. Sci., 65 (1989), 213-220.
%D A002812 J. O. Shallit, Rational numbers with non-terminating, non-periodic modified 
               Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
%H A002812 J. O. Shallit, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/met0.ps">
               Rational numbers with non-terminating, non-periodic modified Engel-type 
               expansions</a>, Fib. Quart., 31 (1993), 37-40.
%H A002812 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NewtonsIteration.html">Newton's Iteration</a>
%H A002812 <a href="Sindx_El.html#Engel">Index entries for sequences related to 
               Engel expansions</a>
%o A002812 (PARI) a(n)=if(n<1,2*(n==0),2*a(n-1)^2-1)
%o A002812 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 
               2008: (Start)
%o A002812 (PARI) /* Roots by recursion. Find first root of ax^2 + b^x + c */
%o A002812 rroot2(a,b,c,p) =
%o A002812 {
%o A002812 local(x=1,x1=1,j);
%o A002812 for(j=1,p,
%o A002812 x1=x;
%o A002812 x=-c/(a*x+b);
%o A002812 x=(x1+x)/2; /* Let x be the average of the last 2 values */
%o A002812 print1(numerator(x)",");
%o A002812 );
%o A002812 }
%o A002812 (End)
%Y A002812 Cf. A071579.
%Y A002812 a(n) = A001075(2^n).
%Y A002812 Cf. A003010.
%Y A002812 Sequence in context: A072059 A102344 A087589 this_sequence A102598 A102747 
               A122524
%Y A002812 Adjacent sequences: A002809 A002810 A002811 this_sequence A002813 A002814 
               A002815
%K A002812 nonn,easy,nice
%O A002812 0,1
%A A002812 N. J. A. Sloane (njas(AT)research.att.com).