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A110090 Numerators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).

%I A110090 #23 Sep 08 2022 08:45:19
%S A110090 0,1,2,2,1,4,4,1,6,6,1,8,8,1,10,10,1,12,12,1,14,14,1,16,16,1,18,18,1,
%T A110090 20,20,1,22,22,1,24,24,1,26,26,1,28,28,1,30,30,1,32,32,1,34,34,1,36,
%U A110090 36,1,38,38,1,40,40,1,42,42,1,44,44,1,46,46,1,48,48,1,50,50,1,52,52,1,54,54
%N A110090 Numerators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).
%C A110090 r(n) = a(n) / A110091(n);
%C A110090 a(n) = (A110091(n-1) + A110091(n-2))/GCD(a(n-1) + a(n-2), A110091(n-1) + A110091(n-2)).
%C A110090 r --> 1.
%H A110090 Vincenzo Librandi, <a href="/A110090/b110090.txt">Table of n, a(n) for n = 0..1000</a>
%H A110090 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).
%F A110090 a(3*k) = 2*k, a(3*k+1) = 1, a(3*k+2) = 2*(k+1).
%F A110090 a(n) = (1/9)*(4*n + 5 + (4*n-4)*cos((2*n+1)*Pi/3) + sqrt(3)*sin(2*(n-1)*Pi/3) - sqrt(3)*sin(4*(n-1)*Pi/3)). - _Wesley Ivan Hurt_, Sep 25 2017
%F A110090 From _Vincenzo Librandi_, Sep 27 2017: (Start)
%F A110090 G.f.: x*(1 +2*x +2*x^2 -x^3)/((1-x)^2*(1+x+x^2)^2).
%F A110090 a(n) = 2*a(n-3) - a(n-6) for n>8. (End)
%e A110090 First terms of r: 0 1 2 2/3 1 4/3 4/5 1 6/5 6/7 1 8/7 ...:
%e A110090 r(2) = (1+1)/(1+0) = 2,   r(3) = (1+1)/(2+1) = 2/3, r(4) = (3+1)/(2+2) = 1,
%e A110090 r(5) = (1+3)/(1+2) = 4/3, r(6) = (3+1)/(4+1) = 4/5, ...
%t A110090 Join[{0,1}, LinearRecurrence[{0,0,2,0,0,-1}, {2,2,1,4,4,1}, 100]] (* or *) CoefficientList[Series[x*(1+2*x+2*x^2-x^3)/((1-x)^2*(1+x+x^2)^2), {x,0,100}], x] (* _Vincenzo Librandi_, Sep 27 2017 *)
%o A110090 (Magma) I:=[0,1,2,2,1,4,4,1]; [n le 8 select I[n] else 2*Self(n-3)-Self(n-6): n in [1..100]]; // _Vincenzo Librandi_, Sep 27 2017
%o A110090 (Sage)
%o A110090 def A110090_list(prec):
%o A110090     P.<x> = PowerSeriesRing(ZZ, prec)
%o A110090     return P( x*(1+2*x+2*x^2-x^3)/(1-x^3)^2 ).list()
%o A110090 A110090_list(100) # _G. C. Greubel_, Jun 16 2021
%Y A110090 Cf. A110091 (denominators).
%K A110090 nonn,easy,frac
%O A110090 0,3
%A A110090 _Reinhard Zumkeller_, Jul 14 2005