A195538 Denominators a(n) of Pythagorean approximations b(n)/a(n) to sqrt(8).

5, 12, 145, 420, 4901, 14280, 166465, 485112, 5654885, 16479540, 192099601, 559819260, 6525731525, 19017375312, 221682772225, 646030941360, 7530688524101, 21946034630940, 255821727047185, 745519146510612, 8690408031080165

Offset: 1

Author

Clark Kimberling, Sep 20 2011

Keywords

• nonn
• frac

Comments

See A195500 for a discussion and references.

Conjecture: a(n) = 35*a(n-2) - 35*a(n-4) + a(n-6) with bisections A098602 and A076218. - R. J. Mathar, Sep 21 2011

Crossrefs

Cf. A195500, A195539, A195540.

Programs

Mathematica

r = Sqrt[8]; 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}]]  (* A195538, A195539 *)
Sqrt[a^2 + b^2] (* A195540 *)
(* Peter J. C. Moses, Sep 02 2011 *)