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

1, 7, 1393, 10812186007, 5055923762956339922096065927393, 516965476521645313412793919264355659075150020437514670946599534039092755401282583412315252007

Offset: 0

Author

Artur Jasinski, Oct 10 2008

Keywords

• nonn

Comments

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

The next term has 279 digits. - Harvey P. Dale, May 31 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..7
• Wikipedia, Halley's method

Crossrefs

Cf. A001333, A006266, A002203, A145452.

Programs

Magma

[Evaluate(DicksonFirst(3^n, -1), 2)/2: n in [0..7]]; // G. C. Greubel, Sep 27 2018; Mar 25 2022

Mathematica

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
NestList[4#^3+3#&,1,5] (* Harvey P. Dale, May 31 2019 *)
LucasL[3^Range[0, 7], 2]/2 (* G. C. Greubel, Mar 25 2022 *)

PARI

A002203(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n;
vector(7, n, n--; A002203(3^n)/2 ) \\ 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

Formula

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.

a(n) = A006266(n)/2.

a(n) = A001333(3^n). - R. J. Mathar, Jan 18 2021