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

Author

Gary W. Adamson, Oct 25 2007

Keywords

• nonn
• easy
• changed

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

Also the number of undirected bishop moves on an n X n chessboard, counted up to rotations of the board. Equivalently, number of undirected bishop moves on an n X n chessboard, counted up to horizontal and vertical reflections of the board. - Hilko Koning, Aug 21 2025

Examples

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

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).

Crossrefs

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

Programs

Magma

[Ceiling((2*n^3+3*n^2+4*n)/12): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011

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 *)
a[n_Integer?Positive] := If[EvenQ[n], With[{m = n/2}, (4 m^3 - 3 m^2 + 2 m)/3],
With[{m = (n - 1)/2}, (4 m^3 + 3 m^2 + 2 m)/3]];
Table[a[n], {n, 1, 50}] (* Hilko Koning, Aug 21 2025 *)

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

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