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

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

Offset: 0

Author

Jeffrey Shallit

Keywords

• nonn
• easy

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]

Crossrefs

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

Programs

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

Mathematica

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

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))

SageMath

[chebyshev_T(2^n, 4) for n in range(11)] # G. C. Greubel, May 17 2023

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