A003487 a(n) = a(n-1)^2 - 2, with a(0) = 5.

5, 23, 527, 277727, 77132286527, 5949389624883225721727, 35395236908668169265765137996816180039862527, 1252822795820745419377249396736955608088527968701950139470082687906021780162741058825727

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The next term has 175 digits. - Harvey P. Dale, Feb 19 2015

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10
• P. Liardet and P. Stambul, Séries d'Engel et fractions continuées, Journal de Théorie des Nombres de Bordeaux 12 (2000), 37-68.
• 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), A003423 (starting with 6). A001601, A145504.

Programs

Maple

a:= n-> simplify(2*ChebyshevT(2^n, 1/2*5), 'ChebyshevT'):
seq(a(n), n=0..7);

Mathematica

NestList[#^2-2&,5,10] (* Harvey P. Dale, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, 2 ChebyshevT[2^n, 5/2]]; (* Michael Somos, Dec 06 2016 *)

PARI

{a(n) = if( n<0, 0, polchebyshev(2^n, 1, 5/2) * 2)}; /* Michael Somos, Dec 06 2016 */

Formula

a(n) = ceiling(c^(2^n)) where c=(5+sqrt(21))/2 is the largest root of x^2-5x+1=0. - Benoit Cloitre, Dec 03 2002

a(n) = 2*T(2^n,5/2) where T(n,x) is the Chebyshev polynomial of the first kind. - Leonid Bedratyuk, Mar 17 2011

Engel expansion of 1/2*(5 - sqrt(21)). Thus 1/2*(5 - sqrt(21)) = 1/5 + 1/(5*23) + 1/(5*23*527) + .... See Liardet and Stambul. Cf. A001566, A003010 and A003423. - Peter Bala, Oct 31 2012

From Peter Bala, Nov 11 2012: (Start)

a(n) = ((5 + sqrt(21))/2)^(2^n) + ((5 - sqrt(21))/2)^(2^n).

sqrt(21)/6 = Product_{n = 0..oo} (1 - 1/a(n)).

sqrt(7/3) = Product_{n = 0..oo} (1 + 2/a(n)).

a(n) - 1 = A145504(n+1). (End)

a(n) = A003501(2^n). - Michael Somos, Dec 06 2016

From Peter Bala, Dec 06 2022: (Start)

a(n) = 2 + 3*Product_{k = 0 ..n-1} (a(k) + 2) for n >= 1.

Let b(n) = a(n) - 5. 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)

Extensions

One more term from Harvey P. Dale, Feb 19 2015