A195556 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/3.

1, 12, 24, 35, 468, 900, 1333, 17760, 34188, 50615, 674424, 1298232, 1922041, 25610340, 49298640, 72986939, 972518508, 1872050076, 2771581645, 36930092952, 71088604260, 105247115567, 1402371013680, 2699494911792, 3996618809905

Offset: 1

Author

Clark Kimberling, Sep 21 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195557, A195558.

Programs

Mathematica

r = 1/3; z = 27;
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}]]  (* A195556, A195557 *)
Sqrt[a^2 + b^2] (* A195558 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Conjecture: a(n) = 37*a(n-3) + 37*a(n-6) - a(n-9). - R. J. Mathar, Sep 21 2011

Empirical g.f.: x*(x^6+12*x^5+24*x^4-2*x^3+24*x^2+12*x+1) / (x^9-37*x^6-37*x^3+1). - Colin Barker, Jun 04 2015