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A005828 a(n) = 2*a(n-1)^2 - 1, a(0) = 4, a(1) = 31.

%I A005828 M3642 #41 May 17 2023 08:42:10
%S A005828 4,31,1921,7380481,108942999582721,23737154316161495960243527681,
%T A005828 1126904990058528673830897031906808442930637286502826475521
%N A005828 a(n) = 2*a(n-1)^2 - 1, a(0) = 4, a(1) = 31.
%C A005828 An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004
%C A005828 The next term has 115 digits. - _Harvey P. Dale_, May 25 2018
%D A005828 Jeffrey Shallit, personal communication.
%D A005828 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005828 G. C. Greubel, <a href="/A005828/b005828.txt">Table of n, a(n) for n = 0..10</a>
%H A005828 Anonymous, <a href="http://www-maths.swan.ac.uk/pgrads/bb/project/node28.html">Fermat's rule for 3-fold perfect numbers</a> [Broken link]
%F A005828 a(n) = A001091(2^n).
%F A005828 From _Peter Bala_, Nov 11 2012, (Start)
%F A005828 a(n) = (1/2)*((4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n)).
%F A005828 2*sqrt(15)/9 = Product_{n>=0} (1 - 1/(2*a(n))).
%F A005828 sqrt(5/3) = Product_{n>=0} (1 + 1/a(n)).
%F A005828 See A002812 for general properties of the recurrence a(n+1) = 2*a(n)^2 - 1.
%F A005828 (End)
%F A005828 a(n) = T(2^n,4), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - _Peter Bala_, Feb 01 2017
%F A005828 a(n) = cos(2^n*arccos(4)). - _Peter Luschny_, Oct 12 2022
%t A005828 NestList[2#^2-1&,4,10] (* _Harvey P. Dale_, May 25 2018 *)
%o A005828 (PARI) a(n)=if(n<1,4*(n==0),2*a(n-1)^2-1)
%o A005828 (PARI) a(n)=if(n<0,0,subst(poltchebi(2^n),x,4))
%o A005828 (Magma) [n le 2 select 2^(3*n-1)-n+1 else 2*Self(n-1)^2 - 1: n in [1..10]]; // _G. C. Greubel_, May 17 2023
%o A005828 (SageMath) [chebyshev_T(2^n, 4) for n in range(11)] # _G. C. Greubel_, May 17 2023
%Y A005828 Cf. A001091, A001601, A002812, A084764 (essentially the same).
%K A005828 nonn,easy
%O A005828 0,1
%A A005828 _Jeffrey Shallit_