A195568 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 7/4.

3, 8, 120, 637, 2176, 30848, 164483, 561288, 7958776, 42435837, 144810240, 2053333248, 10948281603, 37360480520, 529752019320, 2824614217597, 9638859164032, 136673967651200, 728739519858563, 2486788303839624, 35261353901990392

Offset: 1

Author

Clark Kimberling, Sep 21 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195569, A195570.

Programs

Mathematica

r = 7/4; 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}]]  (* A195568, A195569 *)
Sqrt[a^2 + b^2] (* A195570 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Formula

Empirical g.f.: x*(3*x^6+8*x^5+120*x^4-134*x^3+120*x^2+8*x+3) / (x^9-257*x^6-257*x^3+1). - Colin Barker, Jun 04 2015