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

3, 8, 33, 120, 451, 1680, 6273, 23408, 87363, 326040, 1216801, 4541160, 16947843, 63250208, 236052993, 880961760, 3287794051, 12270214440, 45793063713, 170902040408, 637815097923, 2380358351280, 8883618307201, 33154114877520

Offset: 1

Author

Clark Kimberling, Sep 20 2011

Keywords

• nonn
• easy
• frac

Comments

See A195500 for a discussion and references.

Apparently a(n) = A120892(n+1) for 1 <= n <= 24. - Georg Fischer, Oct 24 2018

Examples

			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):
4/3, 15/8, 56/33, 209/120, 780/451.
		

Crossrefs

Cf. A120892, A195500, A195503, A195531.

Programs

Mathematica

r = Sqrt[3]; z = 25;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195499, A195503 *)
Sqrt[a^2 + b^2] (* A195531 *)
(* by Peter J. C. Moses, Sep 02 2011 *)

Formula

Empirical G.f.: x*(3-x)/(1-3*x-3*x^2+x^3). - Colin Barker, Jan 04 2012