cp's OEIS Frontend

A153316 Numerators of continued fraction convergents to sqrt(5/4).

%I A153316 #33 Aug 21 2025 03:31:02
%S A153316 1,9,19,161,341,2889,6119,51841,109801,930249,1970299,16692641,
%T A153316 35355581,299537289,634430159,5374978561,11384387281,96450076809,
%U A153316 204284540899,1730726404001,3665737348901,31056625195209,65778987739319,557288527109761,1180356041958841,10000136862780489
%N A153316 Numerators of continued fraction convergents to sqrt(5/4).
%C A153316 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 A153316 a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)
%C A153316 and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);
%C A153316 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 A153316 and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).
%C A153316 For example, the convergents to sqrt(4/3) start 1/1, 9/8, 19/17, 161/144, 341/305.
%C A153316 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 A153316 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 A153316 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 A153316 for example, if k=4 and n=3, then b(4,n)=a(n) and
%C A153316 4*a(4,6)^2 - a(4,5)*a(4,7) = 4*5473^2 - 2584*46368 = 4;
%C A153316 4*a(4,4)*a(4,6) - a(4,5)^2 = 4*305*5473 - 2584^2 = 4;
%C A153316 b(4,5)*b(4,7) - 4*b(4,6)^2 = 2889*51841 - 4*6119^2 = 5;
%C A153316 b(4,5)^2 - 4*b(4,4)*b(4,6) = 2889^2 - 4*341*6119 = 5.
%H A153316 Colin Barker, <a href="/A153316/b153316.txt">Table of n, a(n) for n = 0..1000</a>
%H A153316 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,18,0,-1).
%F A153316 For n > 0, a(2*n) = 2*a(2*n-1) + a(2*n-2) and a(2*n+1) = 8*a(2*n) + a(2*n-1).
%F A153316 G.f.: (1 + 9*x + x^2 - x^3) / ((1 + 4*x - x^2)*(1 - 4*x - x^2)). - _Colin Barker_, Jan 01 2012
%F A153316 From _Colin Barker_, Mar 27 2016: (Start)
%F A153316 a(n) = ((5*(-2+sqrt(5))^n - 2*sqrt(5)*(-2+sqrt(5))^n + 15*(2+sqrt(5))^n + 6*sqrt(5)*(2+sqrt(5))^n + 3*(2-sqrt(5))^n*(-5+2*sqrt(5)) - (-2-sqrt(5))^n*(5+2*sqrt(5))))/(8*sqrt(5)).
%F A153316 a(n) = 18*a(n-2) - a(n-4) for n > 3. (End)
%F A153316 a(n) = (3 - (-1)^n)*Lucas(3*(n + 1))/8. - _Ehren Metcalfe_, Apr 04 2019
%e A153316 The initial convergents are 1, 9/8, 19/17, 161/144, 341/305, 2889/2584, 6119/5473, 1841/46368, 109801/98209, 930249/832040, 1970299/1762289.
%t A153316 Numerator[Convergents[Sqrt[5/4], 30]] (* _Paolo Xausa_, Jan 16 2025 *)
%o A153316 (PARI) Vec((1+9*x+x^2-x^3)/((1+4*x-x^2)*(1-4*x-x^2)) + O(x^30)) \\ _Colin Barker_, Mar 27 2016
%Y A153316 Cf. A000129, A001333, A142238, A142239, A153315, A153317, A153318.
%K A153316 nonn,easy,changed
%O A153316 0,2
%A A153316 _Charlie Marion_, Jan 07 2009