cp's OEIS Frontend

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

%I A195547 #40 Jul 21 2018 03:18:42
%S A195547 1,4,12,15,80,208,273,1428,3740,4895,25632,67104,87841,459940,1204140,
%T A195547 1576239,8253296,21607408,28284465,148099380,387729212,507544127,
%U A195547 2657535552,6957518400,9107509825,47687540548,124847601996,163427632719,855718194320,2240299317520
%N A195547 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/2.
%C A195547 See A195500 for a discussion and references.
%C A195547 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
%H A195547 Vincenzo Librandi, <a href="/A195547/b195547.txt">Table of n, a(n) for n = 1..1000</a>
%F A195547 a(n) = 2*F(n)*F(n+1)/(2-((n+2)^2 mod 3)), where F(n)=Fibonacci(n). - _Gary Detlefs_, Oct 15 2011
%F A195547 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
%t A195547 r = 1/2; z = 30;
%t A195547 p[{f_, n_}] := (#1[[2]]/#1[[
%t A195547       1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
%t A195547          2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
%t A195547      Array[FromContinuedFraction[
%t A195547         ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
%t A195547 {a, b} = ({Denominator[#1], Numerator[#1]} &)[
%t A195547   p[{r, z}]]  (* A195547, A195548 *)
%t A195547 Sqrt[a^2 + b^2] (* A195549 *)
%t A195547 (* _Peter J. C. Moses_, Sep 02 2011 *)
%t A195547 Table[Numerator[2 Fibonacci[n] Fibonacci[n+1] / ( Fibonacci[n] + Fibonacci[n+1])], {n, 1, 40}] (* _Vincenzo Librandi_, Jul 21 2018 *)
%Y A195547 Cf. A195500, A195548, A195549, A131534.
%K A195547 nonn,frac
%O A195547 1,2
%A A195547 _Clark Kimberling_, Sep 20 2011