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 A097066 Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3). 2
 1, 0, 2, 2, 5, 6, 10, 12, 17, 20, 26, 30, 37, 42, 50, 56, 65, 72, 82, 90, 101, 110, 122, 132, 145, 156, 170, 182, 197, 210, 226, 240, 257, 272, 290, 306, 325, 342, 362, 380, 401, 420, 442, 462, 485, 506, 530, 552, 577, 600, 626, 650, 677, 702, 730, 756, 785, 812 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partial sums of A097065. Pairwise sums are A000124, with extra leading 1. Binomial transform is 1, 1, 3, 9, 26, ..., A072863 with extra leading 1. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). FORMULA G.f.: (1-2*x+2*x^2)/((1-x^2)*(1-x)^2). a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). a(n) = 5*(-1)^n/8 + (2*n^2+3)/8. a(n) = A004652(n+1) - A004526(n+1) = ceiling(((n+1)/2)^2) - floor((n+1)/2). - Ridouane Oudra, Jun 22 2019 E.g.f.: ((4+x+x^2)*cosh(x) - (1-x-x^2)*sinh(x))/4. - G. C. Greubel, Jun 30 2019 MATHEMATICA 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 *) Table[(2n^2 +3 +5(-1)^n)/8, {n, 0, 70}] (* Vincenzo Librandi, Apr 09 2014 *) PROG (PARI) vector(70, n, n--; (2*n^2 +3 +5*(-1)^n)/8) \\ G. C. Greubel, Jun 30 2019 (Magma) [(2*n^2 +3 +5*(-1)^n)/8: n in [0..70]]; // G. C. Greubel, Jun 30 2019 (Sage) [(2*n^2 +3 +5*(-1)^n)/8 for n in (0..70)] # G. C. Greubel, Jun 30 2019 (GAP) List([0..70], n-> (2*n^2 +3 +5*(-1)^n)/8) # G. C. Greubel, Jun 30 2019 CROSSREFS Cf. A000124, A072863, A097065. Cf. A004526, A004652. Sequence in context: A118807 A240309 A098507 * A035548 A240059 A288766 Adjacent sequences: A097063 A097064 A097065 * A097067 A097068 A097069 KEYWORD nonn,easy AUTHOR Paul Barry, Jul 22 2004 STATUS approved

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Last modified December 9 17:12 EST 2022. Contains 358702 sequences. (Running on oeis4.)