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A195499 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).

%I A195499 #27 Oct 24 2018 04:58:25
%S A195499 3,8,33,120,451,1680,6273,23408,87363,326040,1216801,4541160,16947843,
%T A195499 63250208,236052993,880961760,3287794051,12270214440,45793063713,
%U A195499 170902040408,637815097923,2380358351280,8883618307201,33154114877520
%N A195499 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(3).
%C A195499 See A195500 for a discussion and references.
%C A195499 Apparently a(n) = A120892(n+1) for 1 <= n <= 24. - _Georg Fischer_, Oct 24 2018
%F A195499 Empirical G.f.: x*(3-x)/(1-3*x-3*x^2+x^3). - _Colin Barker_, Jan 04 2012
%e A195499 From the Pythagorean triples (3,4,5), (8,15,17),(33,56,65), (120,209,241), (451,780,901), read the first five best approximating fractions b(n)/a(n):
%e A195499 4/3, 15/8, 56/33, 209/120, 780/451.
%t A195499 r = Sqrt[3]; z = 25;
%t A195499 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195499       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195499          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195499      Array[FromContinuedFraction[
%t A195499         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195499 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195499   p[{r, z}]]  (* A195499, A195503 *)
%t A195499 Sqrt[a^2 + b^2] (* A195531 *)
%t A195499 (* by _Peter J. C. Moses_, Sep 02 2011 *)
%Y A195499 Cf. A120892, A195500, A195503, A195531.
%K A195499 nonn,easy,frac
%O A195499 1,1
%A A195499 _Clark Kimberling_, Sep 20 2011