A084764 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=4.

1, 4, 31, 1921, 7380481, 108942999582721, 23737154316161495960243527681, 1126904990058528673830897031906808442930637286502826475521

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jun 04 2003

Keywords

• easy
• nonn

Comments

Product_{k=1..n} (1 + 1/a(k)) converges to sqrt(5/3).

The next term has 115 digits. - Harvey P. Dale, Oct 31 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..11
• H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425.
• Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.

Crossrefs

Cf. A001091, A005828 (essentially the same).

Programs

Magma

[n le 2 select 4^(n-1) else 2*Self(n-1)^2 - 1: n in [1..10]]; // G. C. Greubel, May 16 2023

Mathematica

a[n_]:= a[n]= If[n<2, 4^n, 2 a[n-1]^2 -1]; Table[a[n], {n,0,10}]
Join[{1},NestList[2#^2-1&,4,10]] (* Harvey P. Dale, Oct 31 2013 *)

SageMath

def A084764(n): return 1 if n==0 else chebyshev_T(2^(n-1), 4)
[A084764(n) for n in range(11)] # G. C. Greubel, May 16 2023

Formula

With x=4+sqrt(15), y=4-sqrt(15): a(n+1) = (x^(2^n) + y^(2^n))/2.

a(n) = A005828(n-1), n>0. - R. J. Mathar, Sep 17 2008

a(n) = A001091(2^(n-1)) with a(0) = 1; i.e. a(n) = ChebyshevT(2^(n-1), 4) with a(0) = 1. - G. C. Greubel, May 16 2023