%I A003010 M3494
%S A003010 4,14,194,37634,1416317954,2005956546822746114,
%T A003010 4023861667741036022825635656102100994,
%U A003010 16191462721115671781777559070120513664958590125499158514329308740975788034
%N A003010 A Lucas-Lehmer sequence: a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2.
%C A003010 Albert Beiler states (page 228 of Recreations in the Theory of Numbers): 
               D. H. Lehmer modified Lucas's test to the relatively simple form: 
               If and only if 2^n-1 divides a(n-2) then 2^n-1 is a prime, otherwise 
               it is composite. Since 2^3 - 1 is a factor of a(1) = 14, 2^3 - 1 
               = 7 is a prime. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 07 
               2003
%C A003010 These are the terms in A003500 whose indices are powers of 2. - John 
               Blythe Dobson (j.dobson(AT)uwinnipeg.ca), Oct 28 2007
%D A003010 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003010 Albert Beiler, Recreations in the Theory of Numbers, page 228.
%D A003010 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 A003010 J. S. Hall, A remark on the primeness of Mersenne numbers, J. London 
               Math. Soc. 28, (1953). 285-287.
%H A003010 N. J. A. Sloane, <a href="b003010.txt">Table of n, a(n) for n = 0..10</
               a>
%H A003010 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 A003010 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Lucas-LehmerTest.html">Link to a section of The World of Mathematics.</
               a>
%H A003010 <a href="Sindx_Aa.html#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 
               + ...</a>
%F A003010 a(n) = ceiling((2+sqrt(3))^(2^n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Nov 30 2002
%F A003010 More generally, if u(0)=z, integer>2 and u(n)=a(n-1)^2-2 then u(n)=ceiling(c^(2^n)) 
               where c=(1/2)*(z+sqrt(z^2-4)) is the largest root of x^2-zx+1=0. 
               - Benoit Cloitre, Dec 03, 2002
%F A003010 a(n)=(2+sqrt(3))^(2^(n-1))+(2-sqrt(3))^(2^(n-1)) - John Sillcox (johnsillcox(AT)hotmail.com), 
               Sep 20 2003
%F A003010 a(n) = Ceiling(tan(5*pi/12)^(2^n)) Note: 5*pi/12 radians is 75 degrees 
               - Jason M. Follas (jasonfollas(AT)hotmail.com), Jan 16 2004
%F A003010 Sum_{n>=0} 1/( prod_{k=0..n} a(k) ) = 2-sqrt(3). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Aug 11 2004
%F A003010 Comment from Ulrich Sondermann, Sep 04 2006: To generate the n-th number 
               in the sequence: let x=2^(n-1), a=2, b=sqrt(3). Take every other 
               term of the binomial expansion (a+b)^x times 2.
%F A003010 E.g. For the 4th term: x=2^(4-1)=8, the binomial expansion is: a^8 + 
               7a^7 b + 28a^6 b^2 + 56a^5 b^3 + 70a^4 b^4 + 56a^3 b^5 + 28a^2 b^6 
               + 7a b^7 + b^8, every other term times 2: 2(a^8 + 28a^6 b^2 + 70a^4 
               b^4 + 28a^2 b^6 + b^8) = 2(256 + (28)(64)(3) + (70)(16)(9) + (28)(4)(27) 
               + 81) = 2(18817) = 37634
%F A003010 a(n) = 2*cosh( 2^(n-1)*log(sqrt(3)+2) ) For n>0, a(n) = 2 + 3 * 4^n * 
               product( a(k)/2, k=0..n-2 )^2, where a(k)/2 = A002812(k) is a coprime 
               sequence. - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Mar 09 
               2007
%p A003010 a := n-> if n>0 then a(n-1)^2-2 else 4 fi: 'a(i)' $ i=0..9; - M. F. Hasler 
               (Maximilian.Hasler(AT)gmail.com), Mar 09 2007
%o A003010 (PARI:) a(n)=if(n,a(n-1)^2-2,4) vector(10,i,a(i-1)) - M. F. Hasler (Maximilian.Hasler(AT)gmail.com), 
               Mar 09 2007
%Y A003010 Cf. A001566 (starting with 3), A003423 (starting with 6), A003487 (starting 
               with 5)
%Y A003010 Cf. A002812.
%Y A003010 Sequence in context: A129224 A129225 A129226 this_sequence A118770 A112514 
               A001140
%Y A003010 Adjacent sequences: A003007 A003008 A003009 this_sequence A003011 A003012 
               A003013
%K A003010 nonn
%O A003010 0,1
%A A003010 N. J. A. Sloane (njas(AT)research.att.com).
%E A003010 One more term from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Jan 18 
               2005