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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.

%I A081460 #41 Apr 18 2023 08:30:06
%S A081460 1,4,72,23184,2403763488,25840354427429161536,
%T A081460 2986152136938872067784669198846010266752,
%U A081460 39878504028822311675150039382403961856254569551519724209276629577579916539865344
%N 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.
%C A081460 Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6, etc.
%C A081460 The sequence satisfies the Pell equation A081459(n+1)^2 - 5*a(n+1)^2 = 1. - _Vincenzo Librandi_, Dec 20 2011
%H A081460 Vincenzo Librandi, <a href="/A081460/b081460.txt">Table of n, a(n) for n = 1..11</a>
%F A081460 a(n) = 2*a(n-1)*A081459(n-1). - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
%F A081460 a(n) = A000045(A007283(n-1))/2. - _Ehren Metcalfe_, Oct 07 2017
%F A081460 From _Amiram Eldar_, Apr 07 2023: (Start)
%F A081460 a(n) = A079613(n-1)/2.
%F A081460 a(n) = Product_{k=1..n-1} L(3*2^(n-1-k)), where L(k) is the k-th Lucas number (A000032). (End)
%F A081460 a(n) = A001076(2^(n-1)). - _Robert FERREOL_, Apr 18 2023
%t A081460 Table[Fibonacci[2^(n - 1)*3], {n, 1, 8}]/2 (* _Amiram Eldar_, Apr 07 2023 *)
%o A081460 (PARI) {r=2; N=5; for(n=1,8,a=numerator(r); b=denominator(r); print1(b,","); r=(1/2)*(r + N/r))}
%o A081460 (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
%Y A081460 Cf. A000032, A000045, A000129, A001333, A007283, A079613, A081459.
%K A081460 nonn
%O A081460 1,2
%A A081460 _Amarnath Murthy_, Mar 22 2003
%E A081460 Edited and extended by _Klaus Brockhaus_ and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
%E A081460 a(8) corrected by _Vincenzo Librandi_, Dec 20 2011