%I A001566 M2705 N1084
%S A001566 3,7,47,2207,4870847,23725150497407,562882766124611619513723647,
%T A001566 316837008400094222150776738483768236006420971486980607
%N A001566 a(0) = 3; thereafter, a(n) = a(n-1)^2 - 2.
%C A001566 a(n)=Fibonacci(2^[n+2])/Fibonacci(2^[n+1]) -Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               May 08 2000
%C A001566 Expansion of 1/phi: 1/phi = (1-1/3)*(1-1/((3-1)*7))*(1-1/(((3-1)*7-1)*47))*(1-1/
               ((((3-1)*7-1)*47-1)*2207))... (phi being the golden ration (1+sqrt(5))/
               2) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 06 
               2003
%C A001566 An infinite coprime sequence defined by recursion. - Michael Somos Mar 
               14 2004
%C A001566 Starting with 7, the terms end with 7,47,07,47,07,..., of the form 8a+7 
               where a = 0,1,55,121771,... Conjecture: Every a is square-free, every 
               other a is divisible by 55, the a's are a subset of A046194, the 
               heptagonal triangular numbers (the first,2nd,3rd,6th,11th,?... terms) 
               . - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 08 2004
%C A001566 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 
               2008: (Start)
%C A001566 Also the reduced numerator of the convergents to sqrt(5) using Newton's
%C A001566 recursion x = (5/x+x)/2. (End)
%D A001566 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001566 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001566 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. 397.
%D A001566 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 
               3rd ed., Oxford Univ. Press, 1954, p. 223.
%D A001566 E. Lucas, Nouveaux theoremes d'arithmetique superieure, Comptes Rend., 
               83 (1876), 1286-1288.
%D A001566 M. Mendes France and A. J. van der Poorten, From geometry to Euler identities, 
               Theoret. Comput. Sci., 65 (1989), 213-220.
%H A001566 A. V. Aho and N. J. A. Sloane, <a href="http://www.research.att.com/~njas/
               doc/doubly.html">Some doubly exponential sequences</a>, Fib. Quart., 
               11 (1973), 429-437.
%H A001566 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%F A001566 a(n)=ceiling(c^(2^n)) where c=(3+sqrt(5))/2=tau^2 is the largest root 
               of x^2-3x+1=0. - Benoit Cloitre, Dec 03, 2002
%F A001566 a(n)=Round[c^(2^n)] where c =GoldenRatio=1.6180339887498948482...=(Sqrt[5]+1)/
               2 [From Artur Jasinski (grafix(AT)csl.pl), Sep 22 2008]
%F A001566 a(n)=G^(2^n)+(1-G)^(2^n) = G^(2^n)+(-G)^(-2^n) where G is Golden ratio 
               = (1+Sqrt[5])/2 = 1.618033989 [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 05 2008]
%F A001566 Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008: (Start)
%F A001566 a(n)=A058635(n+1)/A058635(n)
%F A001566 a(n)=(G^(2^(n + 1)) - (1 - G)^(2^(n + 1)))/((G^(2^n)) - (1 - G)^(2^n)) 
               where G = (1 + Sqrt[5])/2 (End)
%F A001566 a(n) = 2*Cosh[2^n*ArcCosh[Sqrt[5]/2] [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 09 2008]
%F A001566 a(n) = Fibonacci(2^(n+1)-1)+Fibonacci(2^(n+1)+1). (3-sqrt5)/2 = 1/3 + 
               1/(3*7) + 1/(3*7*47) + 1/(3*7*47*2207) + ... (E. Lucas) . [From Philippe 
               DELEHAM (kolotoko(AT)wanadoo.fr), Apr 21 2009]
%F A001566 a(n)(a(n+1)-1)/2 = A023039(2^n) [From M. F. Hasler (MHasler(AT)univ-ag.fr), 
               Sep 27 2009]
%e A001566 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 
               2008: (Start)
%e A001566 Init x=1.
%e A001566 x = (5/1+1)/2 = 3/1
%e A001566 x = (5/3+3)/2 = 7/3
%e A001566 x = (5/7/3+7/3)/2 = 47/21
%e A001566 x = (5/47/21+47/21)/2 = 2207/987
%e A001566 (2207/987)^2 = 5.000004106... (End)
%t A001566 c = N[GoldenRatio, 1000]; Table[Round[c^(2^n)], {n, 1, 10}] [From Artur 
               Jasinski (grafix(AT)csl.pl), Sep 22 2008]
%t A001566 c = (1 + Sqrt[5])/2; Table[Expand[c^(2^n) + (-c + 1)^(2^n)], {n, 1, 8}] 
               [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
%t A001566 G = (1 + Sqrt[5])/2; Table[Expand[(G^(2^(n + 1)) - (1 - G)^(2^(n + 1)))/
               Sqrt[5]]/Expand[((G^(2^n)) - (1 - G)^(2^n))/Sqrt[5]], {n, 1, 10}] 
               [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
%t A001566 Table[2*Cosh[2^n*ArcCosh[Sqrt[5]/2],{n,1,30}] [From Artur Jasinski (grafix(AT)csl.pl), 
               Oct 09 2008]
%o A001566 (PARI) a(n)=if(n<1,3*(n==0),a(n-1)^2-2)
%o A001566 Contribution from Cino Hilliard (hillcino368(AT)hotmail.com), Sep 28 
               2008: (Start)
%o A001566 (PARI) g(n,p) = x=1;for(j=1,p,x=(n/x+x)/2;print1(numerator(x)","))
%o A001566 g(5,8) (End)
%Y A001566 Lucas numbers (A000032) with subscripts that are powers of 2 greater 
               than 1 (Herb Wilf). Cf. A000045.
%Y A001566 Cf. A003010 (starting with 4), A003423 (starting with 6), A003487 (starting 
               with 5)
%Y A001566 A058635 [From Artur Jasinski (grafix(AT)csl.pl), Oct 05 2008]
%Y A001566 Sequence in context: A020754 A052381 A031440 this_sequence A019039 A077559 
               A062959
%Y A001566 Adjacent sequences: A001563 A001564 A001565 this_sequence A001567 A001568 
               A001569
%K A001566 easy,nonn,nice
%O A001566 0,1
%A A001566 N. J. A. Sloane (njas(AT)research.att.com).