A081459 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with r = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the numerators.

2, 9, 161, 51841, 5374978561, 57780789062419261441, 6677239169351578707225356193679818792961, 89171045849445921581733341920411050611581102638589828325078491812417901966295041

Offset: 1

Author

Amarnath Murthy, Mar 22 2003

Keywords

• nonn
• easy

Comments

Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6 etc.

The sequence satisfies the Pell equation a(n+1)^2 - 5*A081460(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..11

Crossrefs

Cf. A000032, A000129, A001333, A081460.

Programs

Magma

m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Numerator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011

Mathematica

k = 4; Table[Simplify[Expand[(1/2) (((k + Sqrt[k^2 + 4])/2)^(2^(n - 1)) + ((k - Sqrt[k^2 + 4])/2)^(2^(n - 1)))]], {n, 1, 6}] (* Artur Jasinski, Oct 12 2008 *)
aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (* Artur Jasinski, Oct 12 2008 *)

PARI

{r=2; N=5; for(n=1,8,a=numerator(r); b=denominator(r); print1(a,","); r=(1/2)*(r + N/r) )}

Formula

a(n) = a(n-1)^2 + 5*A081460(n-1)^2. - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003

a(n) = (1/2)*(((4+2*sqrt(5))/2)^(2^(n-1)) + ((4-2*sqrt(5))/2)^(2^(n-1))). a(n+1) = 2*a(n)^2 - 1 for n > 1. - Artur Jasinski, Oct 12 2008

a(n) = A000032(3*2^(n-1))/2. - Ehren Metcalfe, Oct 05 2017

a(n) = A001077(2^(n-1)). - Robert FERREOL, Apr 16 2023

From Peter Bala, Jun 22 2025: (Start)

Product_{n >= 1} (1 + 1/a(n)) = (3/4)*sqrt(5).

Product_{n >= 1} (1 - 1/(2*a(n))) = (6/19)*sqrt(5). See A002812. (End)

Extensions

Edited and extended by Klaus Brockhaus and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003