A005828 a(n) = 2*a(n-1)^2 - 1, a(0) = 4, a(1) = 31. (Formerly M3642)

4, 31, 1921, 7380481, 108942999582721, 23737154316161495960243527681, 1126904990058528673830897031906808442930637286502826475521

Offset

0,1

Comments

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 14 2004

The next term has 115 digits. - Harvey P. Dale, May 25 2018

References

Jeffrey Shallit, personal communication.

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..10

Anonymous, Fermat's rule for 3-fold perfect numbers [Broken link]

Formula

a(n) = A001091(2^n).

From Peter Bala, Nov 11 2012, (Start)

a(n) = (1/2)*((4 + sqrt(15))^(2^n) + (4 - sqrt(15))^(2^n)).

2*sqrt(15)/9 = Product_{n>=0} (1 - 1/(2*a(n))).

sqrt(5/3) = Product_{n>=0} (1 + 1/a(n)).

See A002812 for general properties of the recurrence a(n+1) = 2*a(n)^2 - 1.

(End)

a(n) = T(2^n,4), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Feb 01 2017

a(n) = cos(2^n*arccos(4)). - Peter Luschny, Oct 12 2022

Mathematica

NestList[2#^2-1&, 4, 10] (* Harvey P. Dale, May 25 2018 *)

Prog

(PARI) a(n)=if(n<1, 4*(n==0), 2*a(n-1)^2-1)
(PARI) a(n)=if(n<0, 0, subst(poltchebi(2^n), x, 4))
(Magma) [n le 2 select 2^(3*n-1)-n+1 else 2*Self(n-1)^2 - 1: n in [1..10]]; // G. C. Greubel, May 17 2023
(SageMath) [chebyshev_T(2^n, 4) for n in range(11)] # G. C. Greubel, May 17 2023

Crossrefs

Cf. A001091, A001601, A002812, A084764 (essentially the same).

Sequence in context: A203011 A228467 A005841 * A084764 A061789 A103909

Adjacent sequences: A005825 A005826 A005827 * A005829 A005830 A005831

Keyword

nonn,easy

Author

Jeffrey Shallit

Status

approved