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A003423 a(n) = a(n-1)^2 - 2.
(Formerly M4215)
8
6, 34, 1154, 1331714, 1773462177794, 3145168096065837266706434, 9892082352510403757550172975146702122837936996354 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

If x is either of the roots of x^2 - 6x + 1 = 0 (ie, x = 3 +/- 2sqrt(2)), then x^(2^n) + 1 = a(n)*x^(2^(n-1)). For example, x^8 + 1 = 1154x^4. -_ James East_, Oct 05 2018

REFERENCES

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 376.

E. Lucas, "Théorie des Fonctions Numériques Simplement Périodiques, II", Amer. J. Math., 1 (1878), 289-321.

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10

P. Liardet and P. Stambul, Series d'Engel et fractions continuees, Journal de Théorie des Nombres de Bordeaux 12 (2000), 37-68.

Jeffrey Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211.

J. Shallit, An interesting continued fraction, Math. Mag., 48 (1975), 207-211. [Annotated scanned copy]

J. Shallit & N. J. A. Sloane, Correspondence 1974-1975

Wikipedia, Engel Expansion

Index entries for sequences of form a(n+1)=a(n)^2 + ...

FORMULA

a(n) = ceiling(c^(2^n)) where c = 3 + 2*sqrt(2) is the largest root of x^2 - 6x + 1 = 0. - Benoit Cloitre, Dec 03 2002

From Paul D. Hanna, Aug 11 2004: (Start)

a(n) = (3+sqrt(8))^(2^n) + (3-sqrt(8))^(2^n).

Sum_{n>=0} 1/(Product_{k=0..n} a(k) ) = 3 - sqrt(8). (End)

a(n) = 2*A001601(n+1).

a(n-1) = Round((1 + sqrt(2))^(2^n)). - Artur Jasinski, Sep 25 2008

a(n) = 2*T(2^n,3) where T(n,x) is the Chebyshev polynomial of the first kind. - Leonid Bedratyuk, Mar 17 2011

Engel expansion of 3 - 2*sqrt(2). Thus 3 - 2*sqrt(2) = 1/6 + 1/(6*34) + 1/(6*34*1154) + .... See Liardet and Stambul. - Peter Bala, Oct 31 2012

From Peter Bala, Nov 11 2012: (Start)

4*sqrt(2)/7 = Product_{n = 0..inf} (1 - 1/a(n))

sqrt(2) = Product_{n>=0} (1 + 2/a(n)).

a(n) - 1 = A145505(n+1). (End)

MAPLE

a:= n-> simplify(2*ChebyshevT(2^n, 3), 'ChebyshevT'):

seq(a(n), n=0..7);

MATHEMATICA

a[1] := 6; a[n_] := a[n - 1]^2 - 2; Table[a[n], {n, 1, 8}] (* Stefan Steinerberger, Apr 11 2006 *)

Table[Round[(1 + Sqrt[2])^(2^n)], {n, 1, 7}] (* Artur Jasinski, Sep 25 2008 *)

NestList[#^2-2&, 6, 10] (* Harvey P. Dale, Nov 11 2011 *)

PROG

(PARI) a(n)=if(n<1, 6*(n==0), a(n-1)^2-2)

CROSSREFS

Cf. A001566 (starting with 3), A003010 (starting with 4), A003487 (starting with 5).

Cf. A145505.

Sequence in context: A092336 A262104 A161323 * A230936 A230941 A145000

Adjacent sequences:  A003420 A003421 A003422 * A003424 A003425 A003426

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 18 07:22 EST 2019. Contains 329252 sequences. (Running on oeis4.)