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Denominators a(n) of Pythagorean approximations b(n)/a(n) to 1/2.
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%I #40 Jul 21 2018 03:18:42

%S 1,4,12,15,80,208,273,1428,3740,4895,25632,67104,87841,459940,1204140,

%T 1576239,8253296,21607408,28284465,148099380,387729212,507544127,

%U 2657535552,6957518400,9107509825,47687540548,124847601996,163427632719,855718194320,2240299317520

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

%C See A195500 for a discussion and references.

%C 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 Vincenzo Librandi, <a href="/A195547/b195547.txt">Table of n, a(n) for n = 1..1000</a>

%F 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 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 r = 1/2; z = 30;

%t p[{f_, n_}] := (#1[[2]]/#1[[

%t 1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[

%t 2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[

%t Array[FromContinuedFraction[

%t ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];

%t {a, b} = ({Denominator[#1], Numerator[#1]} &)[

%t p[{r, z}]] (* A195547, A195548 *)

%t Sqrt[a^2 + b^2] (* A195549 *)

%t (* _Peter J. C. Moses_, Sep 02 2011 *)

%t Table[Numerator[2 Fibonacci[n] Fibonacci[n+1] / ( Fibonacci[n] + Fibonacci[n+1])], {n, 1, 40}] (* _Vincenzo Librandi_, Jul 21 2018 *)

%Y Cf. A195500, A195548, A195549, A131534.

%K nonn,frac

%O 1,2

%A _Clark Kimberling_, Sep 20 2011