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%I #24 Sep 08 2022 08:45:14
%S 1,0,2,2,5,6,10,12,17,20,26,30,37,42,50,56,65,72,82,90,101,110,122,
%T 132,145,156,170,182,197,210,226,240,257,272,290,306,325,342,362,380,
%U 401,420,442,462,485,506,530,552,577,600,626,650,677,702,730,756,785,812
%N Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).
%C Partial sums of A097065. Pairwise sums are A000124, with extra leading 1.
%C Binomial transform is 1, 1, 3, 9, 26, ..., A072863 with extra leading 1.
%H Vincenzo Librandi, <a href="/A097066/b097066.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F G.f.: (1-2*x+2*x^2)/((1-x^2)*(1-x)^2).
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F a(n) = 5*(-1)^n/8 + (2*n^2+3)/8.
%F a(n) = A004652(n+1) - A004526(n+1) = ceiling(((n+1)/2)^2) - floor((n+1)/2). - _Ridouane Oudra_, Jun 22 2019
%F E.g.f.: ((4+x+x^2)*cosh(x) - (1-x-x^2)*sinh(x))/4. - _G. C. Greubel_, Jun 30 2019
%t 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 Table[(2n^2 +3 +5(-1)^n)/8, {n,0,70}] (* _Vincenzo Librandi_, Apr 09 2014 *)
%o (PARI) vector(70, n, n--; (2*n^2 +3 +5*(-1)^n)/8) \\ _G. C. Greubel_, Jun 30 2019
%o (Magma) [(2*n^2 +3 +5*(-1)^n)/8: n in [0..70]]; // _G. C. Greubel_, Jun 30 2019
%o (Sage) [(2*n^2 +3 +5*(-1)^n)/8 for n in (0..70)] # _G. C. Greubel_, Jun 30 2019
%o (GAP) List([0..70], n-> (2*n^2 +3 +5*(-1)^n)/8) # _G. C. Greubel_, Jun 30 2019
%Y Cf. A000124, A072863, A097065.
%Y Cf. A004526, A004652.
%K nonn,easy
%O 0,3
%A _Paul Barry_, Jul 22 2004
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