A084765 a(n) = 2*a(n-1)^2 - 1, a(0)=1, a(1)=5.

1, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Product_{k>=1} (1 + 1/a(k)) = sqrt(3/2) (see A010527).

A subsequence of A001079 (cf. formula), which must contain any prime occurring in A001079. The initial term a(0)=1 seems rather unnatural; using the recurrence relation it would yield the constant sequence 1,1,1,... Note that this sequence corresponds to sequence b(n) in Shallit's paper, which starts only at offset n=1. - M. F. Hasler, Sep 27 2009

Since if x is even (x^2-2)/2 = 2*y^2-1 and 10 is even from a(1) onward this is a reduced version of the LL sequence starting with 10 (A135927) as it is reduced by dividing by 2 it is also the difference between two possible LL sequences. - Roderick MacPhee, May 31 2015

For n >= 3, a(n) == 201 (mod 1000) if n is even, a(n) == 801 (mod 1000) if n is odd. - Robert Israel, Jun 01 2015

The next term -- a(8) -- has 128 digits. - Harvey P. Dale, Mar 28 2020

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10
• Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
• H. S. Wilf, Limit of a sequence, Elementary Problem E 1093, Amer. Math. Monthly 61 (1954), 424-425.

Crossrefs

Cf. A001079, A002812, A010527, A084764, A135927.

Programs

Magma

[n le 2 select 5^(n-1) else 2*Self(n-1)^2-1: n in [1..10]]; // Vincenzo Librandi, Jun 02 2015

Maple

1,seq(expand((5+2*sqrt(6))^(2^n)+(5-2*sqrt(6))^(2^n))/2, n=0..10); # Robert Israel, Jun 01 2015

Mathematica

a[n_]:= a[n]= If[n<2, 5^n, 2 a[n-1]^2 -1]; Table[a[n], {n,0,10}]
Join[{1}, NestList[2 #^2 - 1 &, 5, 10]] (* Harvey P. Dale, Mar 28 2020 *)

PARI

first(m)={my(v=[1,5]);for(i=3,m,v=concat(v, 2*v[i-1]^2 - 1));v;} \\ Anders Hellström, Aug 22 2015

SageMath

def A084765(n): return 1 if n==0 else chebyshev_T(2^(n-1), 5)
[A084765(n) for n in range(11)] # G. C. Greubel, May 17 2023

Formula

a(n+1) = (x^(2^n) + y^(2^n))/2, with x = 5 + 2*sqrt(6), y = 5 - 2*sqrt(6).

a(n) = A001079(2^(n-1)) with a(0) = 1. - M. F. Hasler, Sep 27 2009

4*sqrt(6)/11 = Product_{n >= 1} (1 - 1/(2*a(n))). See A002812 for some general properties of the recurrence a(n+1) = 2*a(n)^2 - 1. - Peter Bala, Nov 11 2012

a(n) = cos(2^(n-1)*arccos(5)) for n >= 1. - Peter Luschny, Oct 12 2022