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A145451
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a(n) = (1/2) * ((1 + sqrt(2))^(3^n) + (1 - sqrt(2))^(3^n)).
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3
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = (1/2) * ((1 + sqrt(2))^(3^n) + (1 - sqrt(2))^(3^n)).
a(n+1) = 4*a(n)^3 + 3*a(n), a(0)=1.
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MATHEMATICA
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Table[Simplify[Expand[(1/2) ((1 + Sqrt[2])^(3^n) + (1 - Sqrt[2])^(3^n))]], {n, 0, 5}]
a = {}; k = 1; Do[AppendTo[a, k]; k = 4 k^3 + 3 k, {n, 1, 6}]; a
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PROG
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(PARI)
A002203(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n;
(Magma) [Evaluate(DicksonFirst(3^n, -1), 2)/2: n in [0..7]]; // G. C. Greubel, Sep 27 2018; Mar 25 2022
(Sage) [lucas_number2(3^n, 2, -1)/2 for n in (0..7)] # G. C. Greubel, Mar 25 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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