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%I #31 May 02 2016 00:11:03
%S 0,3,1,2,5,3,4,7,5,6,9,7,8,11,9,10,13,11,12,15,13,14,17,15,16,19,17,
%T 18,21,19,20,23,21,22,25,23,24,27,25,26,29,27,28,31,29,30,33,31,32,35,
%U 33,34,37,35,36,39,37,38,41,39,40,43,41,42,45,43,44,47,45,46,49,47,48,51,49,50,53,51,52,55
%N Expansion of x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)).
%H Ilya Gutkovskiy, <a href="/A272071/a272071.pdf">Illustration</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F O.g.f.: x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)).
%F E.g.f.: 2*(3*x*exp(x) + 3*exp(x) - exp(-x/2)*(3*cos((sqrt(3)*x)/2) - 4*sqrt(3)*sin((sqrt(3)*x)/2)))/9.
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>3.
%F a(n) = 2*(3*n + 4*sqrt(3)*sin((2*Pi*n)/3) - 3*cos((2*Pi*n)/3) + 3)/9.
%F a(n) = 3*n - 2*floor(n/3) - 5*floor((n + 1)/3). - _Vaclav Kotesovec_, Apr 22 2016
%F a(n) mod 2 = A011655(n).
%p A272071:=proc(n) option remember;
%p if n=0 then 0 elif n=1 then 3 elif n=2 then 1 elif n=3 then 2 else
%p a(n-1)+a(n-3)-a(n-4); fi; end: seq(A272071(n), n=0..150); # _Wesley Ivan Hurt_, Apr 20 2016
%t LinearRecurrence[{1, 0, 1, -1}, {0, 3, 1, 2}, 80]
%t Table[2 ((3 n + 4 Sqrt[3] Sin[(2 Pi n)/3] - 3 Cos[(2 Pi n)/3] + 3)/9), {n, 0, 79}]
%t CoefficientList[Series[x*(3 - 2*x + x^2)/((1 - x)^2*(1 + x + x^2)), {x, 0, 100}], x] (* _Wesley Ivan Hurt_, Apr 20 2016 *)
%o (PARI) concat(0, Vec(x*(3-2*x+x^2)/((1-x)^2*(1+x+x^2)) + O(x^99))) \\ _Altug Alkan_, Apr 22 2016
%Y Cf. A011655, A028242.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 19 2016
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