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a(n) = 2*a(n-1)^2 - 1, a(0) = 4, a(1) = 31.
(Formerly M3642)
7

%I M3642 #41 May 17 2023 08:42:10

%S 4,31,1921,7380481,108942999582721,23737154316161495960243527681,

%T 1126904990058528673830897031906808442930637286502826475521

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

%C An infinite coprime sequence defined by recursion. - _Michael Somos_, Mar 14 2004

%C The next term has 115 digits. - _Harvey P. Dale_, May 25 2018

%D Jeffrey Shallit, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A005828/b005828.txt">Table of n, a(n) for n = 0..10</a>

%H 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 a(n) = A001091(2^n).

%F From _Peter Bala_, Nov 11 2012, (Start)

%F a(n) = (1/2)*((4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n)).

%F 2*sqrt(15)/9 = Product_{n>=0} (1 - 1/(2*a(n))).

%F sqrt(5/3) = Product_{n>=0} (1 + 1/a(n)).

%F See A002812 for general properties of the recurrence a(n+1) = 2*a(n)^2 - 1.

%F (End)

%F 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 a(n) = cos(2^n*arccos(4)). - _Peter Luschny_, Oct 12 2022

%t NestList[2#^2-1&,4,10] (* _Harvey P. Dale_, May 25 2018 *)

%o (PARI) a(n)=if(n<1,4*(n==0),2*a(n-1)^2-1)

%o (PARI) a(n)=if(n<0,0,subst(poltchebi(2^n),x,4))

%o (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 (SageMath) [chebyshev_T(2^n, 4) for n in range(11)] # _G. C. Greubel_, May 17 2023

%Y Cf. A001091, A001601, A002812, A084764 (essentially the same).

%K nonn,easy

%O 0,1

%A _Jeffrey Shallit_