A084570 Partial sums of A084263.

1, 2, 6, 12, 23, 38, 60, 88, 125, 170, 226, 292, 371, 462, 568, 688, 825, 978, 1150, 1340, 1551, 1782, 2036, 2312, 2613, 2938, 3290, 3668, 4075, 4510, 4976, 5472, 6001, 6562, 7158, 7788, 8455, 9158, 9900, 10680, 11501, 12362, 13266, 14212, 15203, 16238

Offset: 0

Author

Paul Barry, May 31 2003

Keywords

• easy
• nonn

Comments

Partial sums give A084569.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Programs

Magma

[(-1)^n/4 + (2*n^3+6*n^2+10*n+ 9)/12: n in [0..50]]; // Vincenzo Librandi, Apr 04 2015

Mathematica

LinearRecurrence[{3,-2,-2,3,-1},{1,2,6,12,23},50] (* Harvey P. Dale, Nov 12 2014 *)
CoefficientList[Series[(1 - x + 2 x^2) / ((1 + x) (1 - x)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Apr 04 2015 *)

PARI

a(n) = sum(j=0, n, sum(i=0, j, (i+(-1)^i)));
vector(50, n, n--; a(n)) \\ Michel Marcus, Apr 04 2015

Formula

a(n) = (-1)^n/4 + (2n^3 + 6n^2 + 10n + 9)/12.

a(n) = Sum_{j=0..n} (Sum_{i=0..j} (i + (-1)^i)).

From Arun Giridhar, Apr 03 2015: (Start)

a(n) = ceiling(A006527(n+1) / 2).

a(n) = ceiling((n^3 + 3n^2 + 5n + 3)/6).

(End)

G.f.: (1-x+2*x^2)/((1+x)*(1-x)^4). - Vincenzo Librandi, Apr 04 2015