A195562 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/4.

1, 24, 40, 63, 1600, 2624, 4161, 105560, 173160, 274559, 6965376, 11425920, 18116737, 459609240, 753937576, 1195430079, 30327244480, 49748454080, 78880268481, 2001138526424, 3282644031720, 5204902289663, 132044815499520

Offset: 1

Author

Clark Kimberling, Sep 21 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195563, A195564.

Programs

Mathematica

Remove["Global`*"];
r = 1/4; z = 26;
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}]]  (* A195562, A195563 *)
Sqrt[a^2 + b^2] (* A195564 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Conjecture: a(n) = 65*a(n-3) + 65*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(x^6+24*x^5+40*x^4-2*x^3+40*x^2+24*x+1) / (x^9-65*x^6-65*x^3+1). - Colin Barker, Jun 04 2015