%I A001601 M3042 N1234
%S A001601 1,3,17,577,665857,886731088897,1572584048032918633353217,
%T A001601 4946041176255201878775086487573351061418968498177
%N A001601 a(n) = 2*a(n-1)^2 - 1, if n>1. a(0)=1, a(1)=3.
%C A001601 Reduced numerators of Newton's iteration for sqrt(2). - Eric Weisstein 
               (eric(AT)weisstein.com)
%C A001601 An infinite coprime sequence defined by recursion. - Michael Somos Mar 
               14 2004
%D A001601 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001601 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001601 Problem E1093, Amer. Math. Monthly, 61 (1954), 424-425.
%D A001601 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. 376.
%D A001601 M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, 
               Theoret. Comput. Sci., 65 (1989), 213-220.
%D A001601 J. O. Shallit, Rational numbers with non-terminating, non-periodic modified 
               Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
%H A001601 Dennis Martin, <a href="b001601.txt">Table of n, a(n) for n = 0..11</
               a>
%H A001601 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 A001601 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               NewtonsIteration.html">Link to a section of The World of Mathematics.</
               a>
%H A001601 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SquareRoot.html">Link to a section of The World of Mathematics.</
               a>
%H A001601 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PythagorassConstant.html">Pythagoras's Constant</a>
%H A001601 <a href="Sindx_El.html#Engel">Index entries for sequences related to 
               Engel expansions</a>
%F A001601 For n>0: a(n)=a(n-1)^2+2*A051009(n)^2, a(n)^2=2*A051009(n+1)^2+1. - Mario 
               Catalani (mario.catalani(AT)unito.it), May 27 2003
%F A001601 a(n)=sum(Binomial[2^n, 2r]2^r, r=0, .., 2^(n-1)) - Mario Catalani (mario.catalani(AT)unito.it), 
               May 30 2003
%F A001601 Expansion of 1/sqrt(2) as an infinite product: 1/sqrt(2) = prod(k=1, 
               infinity, 1-1/(a(n)+1)). a(1)=3; a(n) = floor(1/(1-1/(sqrt(2)*prod(k=1, 
               n-1, 1-1/(a(k)+1))))) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), 
               Nov 06 2003
%F A001601 A003423(n)=2*a(n+1).
%F A001601 a(n)=(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n)) [From Artur Jasinski 
               (grafix(AT)csl.pl), Oct 10 2008]
%t A001601 Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n))]], 
               {n, 0, 7}][From Artur Jasinski (grafix(AT)csl.pl), Oct 10 2008]
%o A001601 (PARI) a(n)=if(n<1,n==0,2*a(n-1)^2-1)
%Y A001601 Cf. A051009. a(n) = A001333(2^n).
%Y A001601 Sequence in context: A098138 A009719 A128300 this_sequence A061119 A049985 
               A126579
%Y A001601 Adjacent sequences: A001598 A001599 A001600 this_sequence A001602 A001603 
               A001604
%K A001601 nonn,easy,nice,frac
%O A001601 0,2
%A A001601 N. J. A. Sloane (njas(AT)research.att.com).