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

Author

N. J. A. Sloane, Jun 21 2002

Keywords

• nonn
• easy

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)

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.

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).

Crossrefs

Partial sums of A042968.

Essentially a bisection of A156040.

Programs

Magma

[Ceiling(2*n^2/3): n in [0..100]]; // Wesley Ivan Hurt, Mar 12 2015

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 *)

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

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

Extensions

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

New name from Wesley Ivan Hurt, Mar 13 2015