A097063 Expansion of (1-2*x+3*x^2)/((1+x)*(1-x)^3).

1, 0, 3, 4, 9, 12, 19, 24, 33, 40, 51, 60, 73, 84, 99, 112, 129, 144, 163, 180, 201, 220, 243, 264, 289, 312, 339, 364, 393, 420, 451, 480, 513, 544, 579, 612, 649, 684, 723, 760, 801, 840, 883, 924, 969, 1012, 1059, 1104, 1153, 1200, 1251, 1300, 1353, 1404

Offset: 0

Author

Paul Barry, Jul 22 2004

Keywords

• easy
• nonn

Comments

Partial sums of A097062. Pairwise sums are A002061. Binomial transform is essentially A007466.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

A diagonal of A326296.

Programs

Maple

A097063:=n->(1/4) + (3/4)*(-1)^n + (1/2)*n^2; seq(A097063(n), n=0..50); # Wesley Ivan Hurt, Mar 08 2014

Mathematica

Table[(1/4) + (3/4)*(-1)^n + (1/2)*n^2, {n, 0, 50}] (* Wesley Ivan Hurt, Mar 08 2014 *)

Formula

G.f. : (1-2*x+3*x^2)/((1-x^2)(1-x)^2).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

a(n) = Sum_{k=0..n} (k^2-k+1)*(-1)^(n-k).

a(2n) = A058331(n); a(2n+1) = A046092(n). - R. J. Mathar, Oct 27 2008

a(n) = binomial(n+1, 2) - ceiling((n+1)/2) + 2((n+1) mod 2). - Wesley Ivan Hurt, Mar 08 2014

a(n) = 2*floor(n/2) + ceiling((n-1)^2/2). - M. Ryan Julian Jr., Sep 10 2019

a(n) = A326296(n + 1, n) for n > 0. - Andrew Howroyd, Sep 23 2019