A081460 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with a = 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 denominators.

1, 4, 72, 23184, 2403763488, 25840354427429161536, 2986152136938872067784669198846010266752, 39878504028822311675150039382403961856254569551519724209276629577579916539865344

Offset

1,2

Comments

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

The sequence satisfies the Pell equation A081459(n+1)^2 - 5*a(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011

Links

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

Formula

a(n) = 2*a(n-1)*A081459(n-1). - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003

a(n) = A000045(A007283(n-1))/2. - Ehren Metcalfe, Oct 07 2017

From Amiram Eldar, Apr 07 2023: (Start)

a(n) = A079613(n-1)/2.

a(n) = Product_{k=1..n-1} L(3*2^(n-1-k)), where L(k) is the k-th Lucas number (A000032). (End)

a(n) = A001076(2^(n-1)). - Robert FERREOL, Apr 18 2023

Mathematica

Table[Fibonacci[2^(n - 1)*3], {n, 1, 8}]/2 (* Amiram Eldar, Apr 07 2023 *)

Prog

(PARI) {r=2; N=5; for(n=1, 8, a=numerator(r); b=denominator(r); print1(b, ", "); r=(1/2)*(r + N/r))}
(Magma) m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Denominator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011

Crossrefs

Cf. A000032, A000045, A000129, A001333, A007283, A079613, A081459.

Sequence in context: A344693 A172478 A087315 * A327040 A327112 A284673

Adjacent sequences: A081457 A081458 A081459 * A081461 A081462 A081463

Keyword

nonn

Author

Amarnath Murthy, Mar 22 2003

Extensions

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

a(8) corrected by Vincenzo Librandi, Dec 20 2011

Status

approved