%I A081459
%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.
%F A081459 a(n)=a(n-1)^2+5*A081460(n-1)^2 - Mario Catalani (mario.catalani(AT)unito.it), 
               May 21 2003
%e A081459 a(n)=(1/2)(((4+2Sqrt[5])/2)^(2^(n-1))+((4-2Sqrt[5])/2)^(2^(n-1)) a(n+1)=2*a(n)^2-1 
               and a(1)=9 [From Artur Jasinski (grafix(AT)csl.pl), Oct 12 2008]
%t A081459 Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 12 2008: (Start)
%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}]
%t A081459 or
%t A081459 aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (*Artur 
               Jasinski*) (End)
%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) )}
%Y A081459 Cf. A000129, A001333, A081460.
%Y A081459 Sequence in context: A050995 A117116 A133468 this_sequence A038843 A053294 
               A078524
%Y A081459 Adjacent sequences: A081456 A081457 A081458 this_sequence A081460 A081461 
               A081462
%K A081459 nonn
%O A081459 1,1
%A A081459 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 22 2003
%E A081459 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) 
               and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003