A195544 Denominators a(n) of Pythagorean approximations b(n)/a(n) to Pi.

12, 52, 315, 1044, 3296, 20919, 217620, 450296, 125510644, 138980066871, 289898680472, 3593636117787, 34812833117460, 1934468176818608, 1244148342635075, 86081645453428848, 8659539839551787053788, 138771143651019468539176

Offset: 1

Author

Clark Kimberling, Sep 20 2011

Keywords

• nonn
• frac

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195545, A195546.

Programs

Mathematica

r = Pi; z = 21;
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}]]  (* A195544, A195545 *)
Sqrt[a^2 + b^2] (* A195546 *)
(* Peter J. C. Moses, Sep 02 2011 *)