%I #29 Oct 21 2023 11:50:17

%S 3,11,17,19,23,31,37,39,43,51,57,59,63,71,77,79,83,91,97,99,103,111,

%T 117,119,123,131,137,139,143,151,157,159,163,171,177,179,183,191,197,

%U 199,203,211,217,219,223,231,237,239,243,251,257,259,263,271,277,279,283,291,297,299

%N Expansion of x*(3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)).

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F a(n) = (1/2)*(10*n - (1+2*i)*(-i)^n - (1-2*i)*i^n), where i = sqrt(-1).

%F a(n) = 5*n - 2*sin(Pi*n/2) - cos(Pi*n/2).

%F a(n) = 5*n - A228826(n-1). - _Andrew Howroyd_, Nov 29 2018

%F G.f.: x*(x^3 + x^2 + 5*x + 3) / ((x - 1)^2 *(x^2 + 1)). - _Vincenzo Librandi_, Dec 06 2018

%p seq(coeff(series(x*(x^3+x^2+5*x+3)/((1-x)^2*(1+x^2)),x,n+1), x, n), n = 1 .. 60); # _Muniru A Asiru_, Dec 06 2018

%t CoefficientList[Series[(x^3 + x^2 + 5 x + 3)/((x - 1)^2 (x^2 + 1)), {x, 0, 50}], x] (* or *)

%t a[n_]:= (1/2) (10 n - (1 + 2 * I) (-I)^n - (1 - 2 I) I^n); Simplify[Array[a, 50]] (* _Stefano Spezia_, Nov 29 2018 *)

%t LinearRecurrence[{2, -2, 2, -1}, {3, 11, 17, 19}, 60] (* _Vincenzo Librandi_, Dec 06 2018 *)

%o (PARI) Vec((3 + 5*x + x^2 + x^3)/((1 - x)^2*(1 + x^2)) + O(x^60)) \\ _Andrew Howroyd_, Nov 29 2018

%o (Magma) I:=[3,11,17,19]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..60]]; // _Vincenzo Librandi_, Dec 06 2018

%Y Cf. A228826.

%K nonn,easy

%O 1,1

%A _Mark A. Thomas_, Nov 29 2018