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

1,1

Comments

See A195500 for a discussion and references.

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

Links

Table of n, a(n) for n=1..24.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A120892, A195500, A195503, A195531.

Sequence in context: A148916 A148917 A120892 * A225688 A109655 A184255

Adjacent sequences: A195496 A195497 A195498 * A195500 A195501 A195502

Keyword

nonn,easy,frac

Author

Clark Kimberling, Sep 20 2011

Status

approved