A195687 Denominators a(n) of Pythagorean approximations b(n)/a(n) to (1+sqrt(5))/2 (the golden ratio).

3, 8, 28, 1863, 4400, 433008, 262353, 352207108, 379920428, 18418959496, 91011249895, 978117768540, 11516765628956, 1219780690817560, 708294344602810604, 25852535312829023356, 229222230912132985022679

Offset: 1

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn

Comments

See A195500 for a discussion and references.

Crossrefs

Cf. A195500, A195681, A195682.

Programs

Mathematica

r = GoldenRatio; z = 24;
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}]]  (* A195687, A195688 *)
Sqrt[a^2 + b^2] (* A195689 *)
(* Peter J. C. Moses, Sep 02 2011 *)