A195571 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/5.

1, 40, 60, 99, 4100, 6100, 10101, 418140, 622160, 1030199, 42646200, 63454200, 105070201, 4349494240, 6471706260, 10716130299, 443605766300, 660050584300, 1092940220301, 45243438668340, 67318687892360, 111469186340399, 4614387138404400

Offset: 1

Author

Clark Kimberling, Sep 21 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195572, A195573.

Programs

Mathematica

r = 1/5; 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}]]  (* A195571, A195572 *)
Sqrt[a^2 + b^2] (* A195573 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Conjecture: a(n) = 101*a(n-3) + 101*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(x^6+40*x^5+60*x^4-2*x^3+60*x^2+40*x+1) / (x^9-101*x^6-101*x^3+1). - Colin Barker, Jun 04 2015