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A245581 a(n) = (5 * (1 + (-1)^(1 + n)) + 2 * n^2) / 4.

%I A245581 #23 Apr 14 2023 15:28:57
%S A245581 0,3,2,7,8,15,18,27,32,43,50,63,72,87,98,115,128,147,162,183,200,223,
%T A245581 242,267,288,315,338,367,392,423,450,483,512,547,578,615,648,687,722,
%U A245581 763,800,843,882,927,968,1015,1058,1107,1152,1203,1250,1303,1352,1407
%N A245581 a(n) = (5 * (1 + (-1)^(1 + n)) + 2 * n^2) / 4.
%H A245581 Vincenzo Librandi, <a href="/A245581/b245581.txt">Table of n, a(n) for n = 0..1000</a>
%H A245581 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A245581 a(n) = a(-n).
%F A245581 a(2*n+1) = A097080(n+1).
%F A245581 a(2*n+6) = A245578(2*n+4) = 2*A209350(2*n+4).
%F A245581 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - _Colin Barker_, Jul 26 2014
%F A245581 G.f.: -x*(3*x^2-4*x+3) / ((x-1)^3*(x+1)). - _Colin Barker_, Jul 26 2014
%F A245581 E.g.f.: (exp(x)*x*(x+1) + 5*sinh(x))/2. - _Peter Luschny_, Aug 04 2014
%p A245581 A245581 := n -> (5*(1+(-1)^(1+n))+2*n^2)/4; seq(A245581(n), n=0..53);
%t A245581 CoefficientList[Series[-x (3 x^2 - 4 x + 3)/((x - 1)^3 (x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 27 2014 *)
%o A245581 (Sage)
%o A245581 def A():
%o A245581     a, b, c, d = 0, 3, 2, 7
%o A245581     while True:
%o A245581         yield a
%o A245581         a, b, c, d = b, c, d, a + 2*(d - b)
%o A245581 A245581 = A(); [next(A245581) for n in range(54)]
%o A245581 (PARI) concat(0, Vec(-x*(3*x^2-4*x+3)/((x-1)^3*(x+1)) + O(x^100))) \\ _Colin Barker_, Jul 26 2014
%o A245581 (Magma) [(5*(1+(-1)^(1+n))+2*n^2) / 4: n in [0..60]]; // _Vincenzo Librandi_, Jul 27 2014
%Y A245581 Cf. A097080, A209350, A245578.
%K A245581 nonn,easy
%O A245581 0,2
%A A245581 _Peter Luschny_, Jul 26 2014