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A097066 Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).

%I A097066 #28 Mar 24 2025 22:35:02
%S A097066 1,0,2,2,5,6,10,12,17,20,26,30,37,42,50,56,65,72,82,90,101,110,122,
%T A097066 132,145,156,170,182,197,210,226,240,257,272,290,306,325,342,362,380,
%U A097066 401,420,442,462,485,506,530,552,577,600,626,650,677,702,730,756,785,812
%N A097066 Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).
%C A097066 Partial sums of A097065. Pairwise sums are A000124, with extra leading 1.
%C A097066 Binomial transform is 1, 1, 3, 9, 26, ..., A072863 with extra leading 1.
%H A097066 Vincenzo Librandi, <a href="/A097066/b097066.txt">Table of n, a(n) for n = 0..1000</a>
%H A097066 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A097066 G.f.: (1-2*x+2*x^2)/((1-x^2)*(1-x)^2).
%F A097066 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F A097066 a(n) = 5*(-1)^n/8 + (2*n^2+3)/8.
%F A097066 a(n) = A004652(n+1) - A004526(n+1) = ceiling(((n+1)/2)^2) - floor((n+1)/2). - _Ridouane Oudra_, Jun 22 2019
%F A097066 E.g.f.: ((4+x+x^2)*cosh(x) - (1-x-x^2)*sinh(x))/4. - _G. C. Greubel_, Jun 30 2019
%t A097066 CoefficientList[Series[(1-2x+2x^2)/((1+x)(1-x)^3), {x, 0, 70}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 0, 2, 2}, 70] (* _Harvey P. Dale_, Apr 08 2014 *)
%t A097066 Table[(2n^2 +3 +5(-1)^n)/8, {n,0,70}] (* _Vincenzo Librandi_, Apr 09 2014 *)
%o A097066 (PARI) vector(70, n, n--; (2*n^2 +3 +5*(-1)^n)/8) \\ _G. C. Greubel_, Jun 30 2019
%o A097066 (Magma) [(2*n^2 +3 +5*(-1)^n)/8: n in [0..70]]; // _G. C. Greubel_, Jun 30 2019
%o A097066 (Sage) [(2*n^2 +3 +5*(-1)^n)/8 for n in (0..70)] # _G. C. Greubel_, Jun 30 2019
%o A097066 (GAP) List([0..70], n-> (2*n^2 +3 +5*(-1)^n)/8); # _G. C. Greubel_, Jun 30 2019
%Y A097066 Cf. A000124, A072863, A097065.
%Y A097066 Cf. A004526, A004652.
%K A097066 nonn,easy
%O A097066 0,3
%A A097066 _Paul Barry_, Jul 22 2004