A003487 a(n) = a(n-1)^2 - 2. (Formerly M3926)

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

Offset

0,1

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 + ...

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)

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 *)

Prog

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

Crossrefs

Cf. A001566 (starting with 3), A003010 (starting with 4), A003423 (starting with 6). A001601, A145504.

Sequence in context: A018899 A080990 A172036 * A361402 A055490 A261935

Adjacent sequences: A003484 A003485 A003486 * A003488 A003489 A003490

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved