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A097066
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Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).
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2
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
E.g.f.: ((4+x+x^2)*cosh(x) - (1-x-x^2)*sinh(x))/4. - G. C. Greubel, Jun 30 2019
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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