A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.

8, 136, 2448, 43920, 788120, 14142232, 253772064, 4553754912, 81713816360, 1466294939560, 26311595095728, 472142416783536, 8472251907007928, 152028391909359160, 2728038802461456960, 48952670052396866112, 878420022140682133064

Offset: 1

Author

Clark Kimberling, Sep 22 2011

Keywords

• nonn
• easy

Comments

See A195500 for a discussion and references.

Links

• Colin Barker, Table of n, a(n) for n = 1..797
• Index entries for linear recurrences with constant coefficients, signature (17,17,-1).

Crossrefs

Cf. A007805, A014445, A195500, A195615.

Programs

Mathematica

r = 2; z = 32;
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}]]  (* A195614, A195615 *)
Sqrt[a^2 + b^2] (* A007805 *)
(* Peter J. C. Moses, Sep 02 2011 *)

PARI

Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015

Formula

From Colin Barker, Jun 04 2015: (Start)

G.f.: 8*x / ((x+1)*(x^2-18*x+1)).

a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End)

a(n) = (-4*(-1)^n - (-2+sqrt(5))*(9+4*sqrt(5))^(-n) + (2+sqrt(5))*(9+4*sqrt(5))^n)/10. - Colin Barker, Mar 04 2016

a(n) = A014445(n) * A014445(n+1) / 2. - Diego Rattaggi, Jun 01 2020

a(n) is the numerator of continued fraction [4, ..., 4, 8, 4, ..., 4] with (n-1) 4's before and after the middle 8. - Greg Dresden and Hexuan Wang, Aug 30 2021