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%I M3494 #125 Dec 08 2022 12:39:05
%S 4,14,194,37634,1416317954,2005956546822746114,
%T 4023861667741036022825635656102100994,
%U 16191462721115671781777559070120513664958590125499158514329308740975788034
%N A Lucas-Lehmer sequence: a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2.
%C 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_, Jun 07 2003
%C a(n) - a(n-1) divides a(n+1) - a(n). - _Thomas Ordowski_, Dec 24 2016
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 228.
%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. 399.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A003010/b003010.txt">Table of n, a(n) for n = 0..10</a>
%H A. V. Aho and N. J. A. Sloane, <a href="https://www.fq.math.ca/Scanned/11-4/aho-a.pdf">Some doubly exponential sequences</a>, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437, <a href="http://neilsloane.com/doc/doubly.html">alternative link</a>.
%H Larry Ericksen, <a href="https://web.archive.org/web/20210307224444/http://siauliaims.su.lt/index.php?option=com_content&view=article&id=44&Itemid=9">Primality Testing and Prime Constellations</a>, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008.
%H J. S. Hall, <a href="https://doi.org/10.1112/jlms/s1-28.3.285">A remark on the primeness of Mersenne numbers</a>, J. London Math. Soc. 28, (1953). 285-287.
%H D. H. Lehmer, <a href="https://doi.org/10.1112/jlms/s1-10.2.162">On Lucas's test for the primality of Mersenne's numbers</a>, Journal of the London Mathematical Society 1.3 (1935): 162-165. See page 162.
%H D. H. Lehmer. <a href="/A003009/a003009.pdf">Mathematical Reviews. Review of: Hall, James S. A remark on the primeness of Mersenne numbers. (1953)</a> [Annotated scanned copy]
%H P. Liardet and P. Stambul, <a href="http://www.numdam.org/item?id=JTNB_2000__12_1_37_0">Séries d'Engel et fractions continuées</a>, Journal de Théorie des Nombres de Bordeaux 12 (2000), 37-68.
%H J. Shallit, <a href="/A005248/a005248_1.pdf">An interesting continued fraction</a>, Math. Mag., 48 (1975), 207-211. [Annotated scanned copy]
%H P. Vellucci and A. M. Bersani, <a href="https://arxiv.org/abs/1603.01989">The class of Lucas-Lehmer polynomials</a>, arXiv preprint arXiv:1603.01989 [math.CA], 2016.
%H Pierluigi Vellucci and A. M. Bersani, <a href="https://arxiv.org/abs/1606.09597">New formulas for pi involving infinite nested square roots and Gray code</a>, arXiv preprint arXiv:1606.09597 [math.NT], 2016 (The OEIS is cited in version 1, but has been dropped from version 4.)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucas-LehmerTest.html">Lucas-Lehmer Test.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%H <a href="/index/Aa#AHSL">Index entries for sequences of form a(n+1)=a(n)^2 + ...</a>
%F a(n) = ceiling((2 + sqrt(3))^(2^n)). - _Benoit Cloitre_, Nov 30 2002
%F 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 a(n) = (2+sqrt(3))^(2^n) + (2-sqrt(3))^(2^n). - John Sillcox (johnsillcox(AT)hotmail.com), Sep 20 2003
%F 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 Sum_{n >= 0} 1/( Product_{k = 0..n} a(k) ) = 2 - sqrt(3). - _Paul D. Hanna_, Aug 11 2004
%F From _Ulrich Sondermann_, Sep 04 2006: (Start)
%F 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 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. (End)
%F a(n) = 2*cosh( 2^(n-1)*log(sqrt(3)+2) ) For n > 0, a(n) = 2 + 3 * 4^n * (Product_{k=0..n-2} (a(k)/2))^2, where a(k)/2 = A002812(k) is a coprime sequence. - _M. F. Hasler_, Mar 09 2007
%F a(n) = A003500(2^n). - _John Blythe Dobson_, Oct 28 2007
%F a(n) = 2*T(2^n,2) where T(n,x) is the Chebyshev polynomial of the first kind. - _Leonid Bedratyuk_, Mar 17 2011
%F Engel expansion of 2 - sqrt(3). Thus 2 - sqrt(3) = 1/4 + 1/(4*14) + 1/(4*14*194) + ... as noted by Hanna above. See Liardet and Stambul. Cf. A001566, A003423 and A003487. - _Peter Bala_, Oct 31 2012
%F From _Peter Bala_, Nov 11 2012: (Start)
%F 2*sqrt(3)/5 = Product_{n = 0..oo} (1 - 1/a(n)).
%F sqrt(3) = Product_{n = 0..oo} (1 + 2/a(n)).
%F a(n) - 1 = A145503(n+1).
%F a(n) = 2*A002812(n). (End)
%F a(n+1) - a(n) = a(n)^2 - a(n-1)^2. - _Thomas Ordowski_, Dec 24 2016
%F a(n) = 2*cos(2^n * arccos(2)). - _Ryan Brooks_, Oct 27 2020
%F From _Peter Bala_, Dec 06 2022: (Start)
%F a(n) = 2 + 2*Product_{k = 0..n-1} (a(k) + 2) for n >= 1.
%F Let b(n) = a(n) - 4. The sequence {b(n)} appears to be a strong divisibility sequence, that is, gcd(b(n),b(m)) = b(gcd(n,m)) for n, m >= 1. (End)
%p a := n-> if n>0 then a(n-1)^2-2 else 4 fi: 'a(i)' $ i=0..9; # _M. F. Hasler_, Mar 09 2007
%p a := n-> simplify(2*ChebyshevT(2^n, 2), 'ChebyshevT'): seq(a(n), n=0..7);
%t seqLucasLehmer[0] = 4; seqLucasLehmer[n_] := seqLucasLehmer[n - 1]^2 - 2; Array[seqLucasLehmer, 8, 0] (* _Robert G. Wilson v_, Jun 28 2012 *)
%o (PARI)
%o a(n)=if(n,a(n-1)^2-2,4)
%o vector(10,i,a(i-1)) \\ _M. F. Hasler_, Mar 09 2007
%o (Magma) [n le 1 select 4 else Self(n-1)^2-2: n in [1..10]]; // _Vincenzo Librandi_, Aug 24 2015
%o (Python)
%o from itertools import accumulate
%o def f(anm1, _): return anm1**2 - 2
%o print(list(accumulate([4]*8, f))) # _Michael S. Branicky_, Apr 14 2021
%Y Cf. A001566 (starting with 3), A003423 (starting with 6), A003487 (starting with 5).
%Y Cf. A002812, A145503.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E One more term from Thomas A. Rockwell (LlewkcoRAT(AT)aol.com), Jan 18 2005
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