A097066 Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).

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

Author

Paul Barry, Jul 22 2004

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000124, A072863, A097065.

Cf. A004526, A004652.

Programs

GAP

List([0..70], 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

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 *)

PARI

vector(70, n, n--; (2*n^2 +3 +5*(-1)^n)/8) \\ 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

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