%I M3042 N1234 #114 Jul 09 2023 02:21:10
%S 1,3,17,577,665857,886731088897,1572584048032918633353217,
%T 4946041176255201878775086487573351061418968498177
%N a(n) = 2*a(n-1)^2 - 1, if n>1. a(0)=1, a(1)=3.
%C Reduced numerators of Newton's iteration for sqrt(2). - _Eric W. Weisstein_
%C An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004
%C Evaluation of the 2^n - 1 degree interpolating polynomial of 1/x at Chebyshev nodes in the interval (1,2): v = 1.0; for(i = 1, n, v *= 4*(a(i) - x*v)); v *= 2/a(n+1). - _Jose Hortal_, Apr 07 2012
%C Smallest positive integer x satisfying the Pell equation x^2 - 2^(2*n+1) * y^2 = 1, for n > 0. - _A.H.M. Smeets_, Sep 29 2017
%D 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 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 Dennis Martin, <a href="/A001601/b001601.txt">Table of n, a(n) for n = 0..11</a>
%H Dov Jarden, <a href="/A001602/a001602.pdf">Recurring Sequences</a>, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy]
%H M. Mendes France and A. J. van der Poorten, <a href="http://dx.doi.org/10.1016/0304-3975(89)90045-5">From geometry to Euler identities</a>, Theoret. Comput. Sci., 65 (1989), 213-220.
%H Jeffrey Shallit, <a href="http://www.fq.math.ca/Scanned/31-1/shallit.pdf">Rational numbers with non-terminating, non-periodic modified Engel-type expansions</a>, Fib. Quart., 31 (1993), 37-40.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NewtonsIteration.html">Newtons Iteration</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareRoot.html">Square Root</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagorassConstant.html">Pythagoras's Constant</a>.
%H H. S. Wilf, D. C. B. Marsh and J. V. Whittaker, <a href="http://www.jstor.org/stable/2307914">Problem E1093</a>, Amer. Math. Monthly, 61 (1954), 424-425.
%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%F From Mario Catalani (mario.catalani(AT)unito.it), May 27 2003, May 30 2003: (Start)
%F a(n) = a(n-1)^2 + 2*A051009(n)^2 for n > 0.
%F a(n)^2 = 2*A051009(n+1)^2 + 1.
%F a(n) = Sum_{r=0..2^(n-1)} binomial(2^n, 2*r)*2^r. (End)
%F Expansion of 1/sqrt(2) as an infinite product: 1/sqrt(2) = Product_{k>=1} (1 - 1/(a(n)+1)). a(1)=3; a(n) = floor(1/(1-1/(sqrt(2)*Product_{k=1..n-1} 1-1/(a(k)+1)))). - _Thomas Baruchel_, Nov 06 2003
%F 2*a(n+1) = A003423(n).
%F a(n) = (1/2)*((1 + sqrt(2))^(2^n) + (1 - sqrt(2))^(2^n)). - _Artur Jasinski_, Oct 10 2008
%F For n > 1: a(n) - 1 = 4^n * Product_{i=1..n-2} a(i)^2. - _Jose Hortal_, Apr 13 2012
%F From _Peter Bala_, Nov 11 2012: (Start)
%F 4*sqrt(2)/7 = Product_{n>=1} (1 - 1/(2*a(n))).
%F sqrt(2) = Product_{n>=1} (1 + 1/a(n)).
%F a(n) = (1/2)*A003423(n-1). (End)
%F a(n) = cos(2^(n-1) * arccos(3)) = cosh(2^(n-1) * log(3 + 2*sqrt(2))) for n >= 1. - _Daniel Suteu_, Jul 28 2016
%F a(n+1) = T(2^n,3), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Feb 01 2017
%F a(n) = A001541(2^(n-1)). - _A.H.M. Smeets_, May 28 2017
%t Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(2^n) + (1 - Sqrt[2])^(2^n))]], {n, 0, 7}] (* _Artur Jasinski_, Oct 10 2008 *)
%t Join[{1},NestList[2#^2-1&,3,7]] (* _Harvey P. Dale_, Mar 24 2011 *)
%o (PARI) a(n)=if(n<1,n==0,2*a(n-1)^2-1)
%Y Cf. A051009, A003423.
%Y a(n) = A001333(2^n).
%K nonn,easy,nice,frac
%O 0,2
%A _N. J. A. Sloane_