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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
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.
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
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Jul 22 2004
STATUS
approved