Search: id:A142239
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%I A142239
%S A142239 1,4,9,40,89,396,881,3920,8721,38804,86329,384120,854569,3802396,8459361,
%T A142239 37639840,83739041,372596004,828931049,3688320200,8205571449,36510605996,
%U A142239 81226783441,361417739760,804062262961,3577666791604,7959395846169,35415250176280
%N A142239 Denominators of continued fraction convergents to sqrt(3/2).
%C A142239 sqrt(3/2) = 1.224744871... = 2/2 + 2/9 + 2/(9*89) + 2/(89*881) + 2/(881*8721) 
               + 2/(8721*86329), + ... [From /Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 08 2008]
%C A142239 Contribution from Charlie Marion (charliemath(AT)optonline.net), Jan 
               07 2009: (Start)
%C A142239 In general, denominators, a(k,n) and numerators, b(k,n), of continued
%C A142239 fraction convergents to sqrt((k+1)/k) may be found as follows:
%C A142239 a(k,0) = 1, a(k,1) = 2k; for n>0, a(k,2n) = 2*a(k,2n-1)+a(k,2n-2)
%C A142239 and a(k,2n+1)=(2k)*a(k,2n)+a(k,2n-1);
%C A142239 b(k,0) = 1, b(k,1) = 2k+1; for n>0, b(k,2n) = 2*b(k,2n-1)+b(k,2n-2)
%C A142239 and b(k,2n+1)=(2k)*b(k,2n)+b(k,2n-1).
%C A142239 For example, the convergents to sqrt(3/2) start 1/1, 5/4, 11/9,
%C A142239 49/40, 109/89.
%C A142239 In general, if a(k,n) and b(k,n) are the denominators and numerators,
%C A142239 respectively, of continued fraction convergents to sqrt((k+1)/k)
%C A142239 as defined above, then
%C A142239 k*a(k,2n)^2-a(k,2n-1)*a(k,2n+1)=k=k*a(k,2n-2)*a(k,2n)-a(k,2n-1)^2 and
%C A142239 b(k,2n-1)*b(k,2n+1)-k*b(k,2n)^2=k+1=b(k,2n-1)^2-k*b(k,2n-2)*b(k,2n);
%C A142239 for example, if k=2 and n=3, then a(2,n)=a(n) and
%C A142239 2*a(2,6)^2-a(2,5)*a(2,7)=2*881^2-396*3920=2;
%C A142239 2*a(2,4)*a(2,6)-a(2,5)^2=2*89*881-396^2=2;
%C A142239 b(2,5)*b(2,7)-2*b(2,6)^2=485*4801-2*1079^2=3;
%C A142239 b(2,5)^2-2*b(2,4)*b(2,6)=485^2-2*109*1079=3.
%C A142239 Cf. A000129, A001333, A142238, A153313-153318.
%C A142239 [From Charlie Marion (charliemath(AT)optonline.net), Jan 07 2009]
%C A142239 (End)
%F A142239 G.f.'s for numerators and denominators are -(1+5*x+x^2-x^3)/(-1-x^4+10*x^2) 
               and -(1+4*x-x^2)/(-1-x^4+10*x^2).
%e A142239 The initial convergents are 1, 5/4, 11/9, 49/40, 109/89, 485/396, 1079/
               881, 4801/3920, 10681/8721, 47525/38804, 105731/86329, ...
%p A142239 with(numtheory): cf := cfrac (sqrt(3)/sqrt(2),100): [seq(nthnumer(cf,
               i), i=0..50)]; [seq(nthdenom(cf,i), i=0..50)]; [seq(nthconver(cf,
               i), i=0..50)];
%Y A142239 Cf. A115754, A142238.
%Y A142239 Sequence in context: A149156 A149157 A149158 this_sequence A118639 A149159 
               A149160
%Y A142239 Adjacent sequences: A142236 A142237 A142238 this_sequence A142240 A142241 
               A142242
%K A142239 nonn
%O A142239 0,2
%A A142239 N. J. A. Sloane (njas(AT)research.att.com), Oct 05 2008, following a 
               suggestion from Rob Miller (rmiller(AT)AmtechSoftware.net)

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