A195499 - OEIS

%I #27 Oct 24 2018 04:58:25

%S 3,8,33,120,451,1680,6273,23408,87363,326040,1216801,4541160,16947843,

%T 63250208,236052993,880961760,3287794051,12270214440,45793063713,

%U 170902040408,637815097923,2380358351280,8883618307201,33154114877520

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

%C See A195500 for a discussion and references.

%C Apparently a(n) = A120892(n+1) for 1 <= n <= 24. - _Georg Fischer_, Oct 24 2018

%F Empirical G.f.: x*(3-x)/(1-3*x-3*x^2+x^3). - _Colin Barker_, Jan 04 2012

%e 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 4/3, 15/8, 56/33, 209/120, 780/451.

%t r = Sqrt[3]; z = 25;

%t p[{f_, n_}] := (#1[[2]]/#1[[

%t       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[

%t          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[

%t      Array[FromContinuedFraction[

%t         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];

%t {a, b} = ({Denominator[#1], Numerator[#1]} &)[

%t   p[{r, z}]]  (* A195499, A195503 *)

%t Sqrt[a^2 + b^2] (* A195531 *)

%t (* by _Peter J. C. Moses_, Sep 02 2011 *)

%Y Cf. A120892, A195500, A195503, A195531.

%K nonn,easy,frac

%O 1,1

%A _Clark Kimberling_, Sep 20 2011