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A145451 a(n) = (1/2) * ((1 + sqrt(2))^(3^n) + (1 - sqrt(2))^(3^n)).

%I A145451 #41 Sep 08 2022 08:45:38
%S A145451 1,7,1393,10812186007,5055923762956339922096065927393,
%T A145451 516965476521645313412793919264355659075150020437514670946599534039092755401282583412315252007
%N A145451 a(n) = (1/2) * ((1 + sqrt(2))^(3^n) + (1 - sqrt(2))^(3^n)).
%C A145451 Empirical observation: a(n) is the numerator of the lowest-terms fraction representing the n-th approximation of sqrt(2) that is obtained via Halley's method when finding the root of x^2 - 2 = 0, starting with x=1 for n=0.  Halley's method gives the next value of x as x * (x^2 + 6) / (3*x^2 + 2). - _Lee A. Newberg_, Apr 27 2018
%C A145451 The next term has 279 digits. - _Harvey P. Dale_, May 31 2019
%H A145451 G. C. Greubel, <a href="/A145451/b145451.txt">Table of n, a(n) for n = 0..7</a>
%H A145451 Wikipedia, <a href="https://en.wikipedia.org/wiki/Halley%27s_method">Halley's method</a>
%F A145451 a(n) = (1/2) * ((1 + sqrt(2))^(3^n) + (1 - sqrt(2))^(3^n)).
%F A145451 a(n+1) = 4*a(n)^3 + 3*a(n), a(0)=1.
%F A145451 a(n) = A006266(n)/2.
%F A145451 a(n) = A001333(3^n). - _R. J. Mathar_, Jan 18 2021
%t A145451 Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(3^n) + (1 - Sqrt[2])^(3^n))]], {n, 0, 5}]
%t A145451 a = {}; k = 1; Do[AppendTo[a, k]; k = 4 k^3 + 3 k, {n, 1, 6}]; a
%t A145451 NestList[4#^3+3#&,1,5] (* _Harvey P. Dale_, May 31 2019 *)
%t A145451 LucasL[3^Range[0, 7], 2]/2 (* _G. C. Greubel_, Mar 25 2022 *)
%o A145451 (PARI)
%o A145451 A002203(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n;
%o A145451 vector(7, n, n--; A002203(3^n)/2 ) \\ _G. C. Greubel_, Sep 27 2018; Mar 25 2022
%o A145451 (Magma) [Evaluate(DicksonFirst(3^n, -1), 2)/2: n in [0..7]]; // _G. C. Greubel_, Sep 27 2018; Mar 25 2022
%o A145451 (Sage) [lucas_number2(3^n,2,-1)/2 for n in (0..7)] # _G. C. Greubel_, Mar 25 2022
%Y A145451 Cf. A001333, A006266, A002203, A145452.
%K A145451 nonn
%O A145451 0,2
%A A145451 _Artur Jasinski_, Oct 10 2008