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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 April 15 04:01 EDT 2021. Contains 342974 sequences. (Running on oeis4.)