A195559 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2/3.

1, 4, 91, 240, 840, 989, 5396, 117719, 312120, 1089720, 1284121, 7003804, 152799571, 405130920, 1414456320, 1666787669, 9090932396, 198333725039, 525859622640, 1835963213040, 2163489110641, 11800023246004, 257437022301451

Offset: 1

Author

Clark Kimberling, Sep 21 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195560, A195561.

Programs

Mathematica

r = 2/3; 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}]]  (* A195559, A195560 *)
Sqrt[a^2 + b^2] (* A195561 *)
(* Peter J. C. Moses, Sep 02 2011 *)