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A153315 Denominators of continued fraction convergents to sqrt(5/4).

%I A153315 #33 Jan 16 2025 08:26:22
%S A153315 1,8,17,144,305,2584,5473,46368,98209,832040,1762289,14930352,
%T A153315 31622993,267914296,567451585,4807526976,10182505537,86267571272,
%U A153315 182717648081,1548008755920,3278735159921,27777890035288,58834515230497,498454011879264,1055742538989025,8944394323791464
%N A153315 Denominators of continued fraction convergents to sqrt(5/4).
%C A153315 In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:
%C A153315 a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)
%C A153315 and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);
%C A153315 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 A153315 and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).
%C A153315 For example, the convergents to sqrt(4/3) start 1/1, 9/8, 19/17, 161/144, 341/305.
%C A153315 In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then
%C A153315 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 A153315 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 A153315 for example, if k=4 and n=3, then a(4,n)=a(n) and
%C A153315 4*a(4,6)^2 - a(4,5)*a(4,7) = 4*5473^2 - 2584*46368 = 4;
%C A153315 4*a(4,4)*a(4,6) - a(4,5)^2 = 4*305*5473 - 2584^2 = 4;
%C A153315 b(4,5)*b(4,7) - 4*b(4,6)^2 = 2889*51841 - 4*6119^2 = 5;
%C A153315 b(4,5)^2 - 4*b(4,4)*b(4,6) = 2889^2 - 4*341*6119 = 5.
%H A153315 Paolo Xausa, <a href="/A153315/b153315.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H A153315 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,18,0,-1).
%F A153315 For n > 0, a(2n) = 2a(2n-1) + a(2n-2) and a(2n+1) = 8a(2n) + a(2n-1).
%F A153315 Empirical g.f.: (1 + 8*x - x^2)/(1 - 18*x^2 + x^4). - _Colin Barker_, Jan 01 2012
%F A153315 a(n) = (3 - (-1)^n)*Fibonacci(3*(n + 1))/4. - _Ehren Metcalfe_, Apr 04 2019
%e A153315 The initial convergents are 1, 9/8, 19/17, 161/144, 341/305, 2889/2584, 6119/5473, 51841/46368, 109801/98209, 930249/832040, 1970299/1762289, ...
%t A153315 Denominator[Convergents[Sqrt[5/4], 30]] (* _Harvey P. Dale_, Aug 17 2012 *)
%Y A153315 Cf. A000129, A001333, A142238, A142239, A153316, A153317, A153318.
%K A153315 nonn,frac,easy
%O A153315 0,2
%A A153315 _Charlie Marion_, Jan 07 2009
%E A153315 Corrected and extended by _Harvey P. Dale_, Aug 17 2012