A131941 Partial sums of ceiling(n^2/2) (A000982).

0, 1, 3, 8, 16, 29, 47, 72, 104, 145, 195, 256, 328, 413, 511, 624, 752, 897, 1059, 1240, 1440, 1661, 1903, 2168, 2456, 2769, 3107, 3472, 3864, 4285, 4735, 5216, 5728, 6273, 6851, 7464, 8112, 8797, 9519, 10280, 11080, 11921, 12803, 13728, 14696, 15709

Offset

0,3

Comments

Binomial transform of [0, 1, 1, 2, -2, 4, -8, 16, -32, ...].

Starting with offset 1 = (1, 3, 5, 7, ...) convolved with (1, 0, 3, 0, 5, ...). - Gary W. Adamson, Feb 16 2009

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1, p. 11.

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

Formula

For even n, a(n) = n*(2*n^2 +3*n +4)/12. For odd n, a(n) = (n+1)*(2*n^2 +n +3)/12. - Washington Bomfim, Jul 31 2008

From R. J. Mathar, Feb 24 2010: (Start)

G.f.: x*(1+x^2)/((1+x)*(1-x)^4).

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)

From Mircea Merca, Oct 10 2010: (Start)

a(n) = round((2*n^3 + 3*n^2 + 4*n)/12) = round((2*n+1)*(2*n^2 + 3*n + 3)/24) = floor((n+1)*(2*n^2 + n + 3)/12) = ceiling((2*n^3 + 3*n^2 + 4*n)/12).

a(n) = a(n-2) + n^2 - n + 1, n > 1. (End)

a(n) = (2*n*(2*n^2 + 3*n + 4) - 3*(-1)^n + 3)/24. - Bruno Berselli, Dec 07 2010

E.g.f.: (x*(9 + 9*x + 2*x^2)*cosh(x) + (3 + 9*x + 9*x^2 + 2*x^3)*sinh(x))/12. - Stefano Spezia, Dec 21 2021

Example

a(3) = 8 = 0 + 1 + 2 + 5.

Maple

a(n):=round(1/(12)(2*n^(3)+3*n^(2)+4*n))  # Mircea Merca, Oct 10 2010

Mathematica

CoefficientList[Series[x (1 + x^2)/(1 + x)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)

Prog

(PARI) a(n) = (n+[0, 1][n%2+1]) * (2*n^2 +[3, 1][n%2+1]*n +[4, 3][n%2+1])/12 \\ Washington Bomfim, Jul 31 2008
(Magma) [Ceiling((2*n^3+3*n^2+4*n)/12): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

Crossrefs

Cf. A000982, A080930 (binomial transform without leading 0).

Sequence in context: A225253 A254875 A025202 * A009858 A169947 A167616

Adjacent sequences: A131938 A131939 A131940 * A131942 A131943 A131944

Keyword

nonn,easy

Author

Gary W. Adamson, Oct 25 2007

Status

approved