A071619 a(n) = ceiling(2*n^2 / 3).

0, 1, 3, 6, 11, 17, 24, 33, 43, 54, 67, 81, 96, 113, 131, 150, 171, 193, 216, 241, 267, 294, 323, 353, 384, 417, 451, 486, 523, 561, 600, 641, 683, 726, 771, 817, 864, 913, 963, 1014, 1067, 1121, 1176, 1233, 1291, 1350, 1411, 1473, 1536, 1601, 1667, 1734, 1803, 1873

Offset

0,3

Comments

Old name was: If n mod 3 = 0 then 6*(n/3)^2 elif n mod 3 = 1 then 6*((n-2)/3)^2+8*(n-2)/3 + 3 else 6*((n-1)/3)^2+4*(n-1)/3+1.

From Ray G. Opao, Oct 20 2022: (Start)

For n >= 3, a(n) is the maximum number of objects that can be placed on an n X n grid such that no 3 adjacent grid points on the same row or column are occupied.

The first 5 terms of this description are illustrated in the Opao link. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K. M. Tamizhmani, Blending two discrete integrability criteria: singularity confinement and algebraic entropy, arXiv:nlin/0104020 [nlin.SI], 2001.

Ray G. Opao, Illustration for first 5 terms of comment

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

Formula

From Vladeta Jovovic, Jun 23 2002: (Start)

a(n) = (2/3)*n^2 if n mod 3 = 0, otherwise (2/3)*n^2 + 1/3.

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

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

a(n) = ceiling(2*n^2 / 3). - Wesley Ivan Hurt, Jun 20 2013

a(n) + a(n+1) + a(n+2) = A005893(n+1). - R. J. Mathar, Mar 01 2014

a(n+1) = A156040(2*n). - L. Edson Jeffery, Jul 30 2014

Let F(x,y) = 6*((x-y)/3)^2 + 4*y*(x-y)/3 + y*(y+1)/2; then a(n) = F(n,(n mod 3)). - R. J. Cano, Jul 30 2014

a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n)/3). - Wesley Ivan Hurt, Mar 12 2015

a(n) = n^2 - floor(n^2/3) = (2/9)*(3*n^2 + 1 - cos(2*Pi*n/3)). - Bruno Berselli, Jan 18 2017

E.g.f.: (2*exp(x)*(1 + 3*x*(1 + x)) - 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022

Sum_{n>=1} 1/a(n) = Pi^2/36 + 3*c*sinh(c)/(1+2*cosh(c)), where c = Pi*sqrt(2)/3. - Amiram Eldar, Jan 08 2023

Maple

A071619 := proc(n) if n mod 3 = 0 then 6*(n/3)^2 elif n mod 3 = 1 then 6*((n-1)/3)^2+4*(n-1)/3+1 else 6*((n-2)/3)^2+8*(n-2)/3 +3; fi; end;

Mathematica

f[n_] := Which[Divisible[n, 3], 6(n/3)^2, Mod[n, 3] == 1, 6(((n - 1)/3)^2) + 4 (n - 1)/3 + 1, True, 6((n - 2)/3)^2 + 8((n - 2)/3) + 3]; Array[f, 60, 0] (* or *) LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 3, 6, 11}, 60] (* Harvey P. Dale, Feb 28 2012 *)
CoefficientList[Series[((x * (1 + x) * (1 + x^2))/((1 + x + x^2) * (1 - x)^3)), {x, 0, 53}], x] (* L. Edson Jeffery, Jul 30 2014 *)
Ceiling[2Range[0, 49]^2/3] (* Alonso del Arte, Mar 13 2015 *)
Table[n^2 - Floor[n^2/3], {n, 0, 60}] (* Bruno Berselli, Jan 18 2017 *)

Prog

(PARI) f=(x, y)->6*((x-y)/3)^2+4*y*(x-y)/3+y*(y+1)/2;
a(n)=f(n, n%3); \\ R. J. Cano, Jul 20 2014
(Magma) [Ceiling(2*n^2/3): n in [0..100]]; // Wesley Ivan Hurt, Mar 12 2015

Crossrefs

Partial sums of A042968.

Essentially a bisection of A156040.

Sequence in context: A174648 A178419 A065504 * A025735 A023601 A173143

Adjacent sequences: A071616 A071617 A071618 * A071620 A071621 A071622

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 21 2002

Extensions

Corrected definition (Old Name) from Harvey P. Dale, Feb 28 2012

New name from Wesley Ivan Hurt, Mar 13 2015

Status

approved