A195547 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/2.

1, 4, 12, 15, 80, 208, 273, 1428, 3740, 4895, 25632, 67104, 87841, 459940, 1204140, 1576239, 8253296, 21607408, 28284465, 148099380, 387729212, 507544127, 2657535552, 6957518400, 9107509825, 47687540548, 124847601996, 163427632719, 855718194320, 2240299317520

Offset: 1

Author

Clark Kimberling, Sep 20 2011

Keywords

• nonn
• frac

Comments

See A195500 for a discussion and references.

a(n) is the numerator of the harmonic mean of F(n) and F(n+1), where F = A000045 (Fibonacci numbers). Example: 2*F(9)*F(10)/(F(9)+F(10)) = 2*34*55/(34+55) = 3740/89, therefore a(9) = 3740. - Francesco Daddi, Jul 04 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A195500, A195548, A195549, A131534.

Programs

Mathematica

r = 1/2; z = 30;
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}]]  (* A195547, A195548 *)
Sqrt[a^2 + b^2] (* A195549 *)
(* Peter J. C. Moses, Sep 02 2011 *)
Table[Numerator[2 Fibonacci[n] Fibonacci[n+1] / ( Fibonacci[n] + Fibonacci[n+1])], {n, 1, 40}] (* Vincenzo Librandi, Jul 21 2018 *)

Formula

a(n) = 2*F(n)*F(n+1)/(2-((n+2)^2 mod 3)), where F(n)=Fibonacci(n). - Gary Detlefs, Oct 15 2011

Empirical G.f.: x*(1+4*x+12*x^2-2*x^3+12*x^4+4*x^5+x^6)/(1-17*x^3-17*x^6+x^9). - Colin Barker, Apr 15 2012