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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.

%I A081459 #44 Jun 23 2025 10:23:34
%S A081459 2,9,161,51841,5374978561,57780789062419261441,
%T A081459 6677239169351578707225356193679818792961,
%U A081459 89171045849445921581733341920411050611581102638589828325078491812417901966295041
%N 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.
%C A081459 Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6 etc.
%C A081459 The sequence satisfies the Pell equation a(n+1)^2 - 5*A081460(n+1)^2 = 1. - _Vincenzo Librandi_, Dec 20 2011
%H A081459 Vincenzo Librandi, <a href="/A081459/b081459.txt">Table of n, a(n) for n = 1..11</a>
%F A081459 a(n) = a(n-1)^2 + 5*A081460(n-1)^2. - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
%F A081459 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
%F A081459 a(n) = A000032(3*2^(n-1))/2. - _Ehren Metcalfe_, Oct 05 2017
%F A081459 a(n) = A001077(2^(n-1)). - _Robert FERREOL_, Apr 16 2023
%F A081459 From  _Peter Bala_, Jun 22 2025: (Start)
%F A081459 Product_{n >= 1} (1 + 1/a(n)) = (3/4)*sqrt(5).
%F A081459 Product_{n >= 1} (1 - 1/(2*a(n))) = (6/19)*sqrt(5). See A002812. (End)
%t A081459 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 *)
%t A081459 aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (* _Artur Jasinski_, Oct 12 2008 *)
%o A081459 (PARI) {r=2; N=5; for(n=1,8,a=numerator(r); b=denominator(r); print1(a,","); r=(1/2)*(r + N/r) )}
%o A081459 (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
%Y A081459 Cf. A000032, A000129, A001333, A081460.
%K A081459 nonn,easy
%O A081459 1,1
%A A081459 _Amarnath Murthy_, Mar 22 2003
%E A081459 Edited and extended by _Klaus Brockhaus_ and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003