A003423 a(n) = a(n-1)^2 - 2, with a(0) = 6.

6, 34, 1154, 1331714, 1773462177794, 3145168096065837266706434, 9892082352510403757550172975146702122837936996354

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If x is either of the roots of x^2 - 6*x + 1 = 0 (i.e., x = 3 +- 2*sqrt(2)), then x^(2^n) + 1 = a(n)*x^(2^(n-1)). For example, x^8 + 1 = 1154*x^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.
• 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.
• E. Lucas, Théorie des Fonctions Numériques Simplement Périodiques, II, Amer. J. Math., 1 (1878), 289-321.
• 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 + ...

Crossrefs

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

Cf. A145505.

Programs

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 *)

PARI

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

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} (1 - 1/a(n))

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

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

From Peter Bala, Dec 06 2022: (Start)

a(n) = 2 + 4*Product_{k = 0 ..n-1} (a(k) + 2) for n >= 1.

Let b(n) = a(n) - 6. The sequence {b(n)} appears to be a strong divisibility sequence, that is, gcd(b(n),b(m)) = b(gcd(n,m)) for n, m >= 1. (End)