A002264 Nonnegative integers repeated 3 times.

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of A010872, since A010872(n) + 3*a(n) = n. - Hieronymus Fischer, Jun 01 2007

Chvátal proved that, given an arbitrary n-gon, there exist a(n) points such that all points in the interior are visible from at least one of those points; further, for all n >= 3, there exists an n-gon which cannot be covered in this fashion with fewer than a(n) points. This is known as the "art gallery problem". - Charles R Greathouse IV, Aug 29 2012

The inverse binomial transform is 0, 0, 0, 1, -3, 6, -9, 9, 0, -27, 81, -162, 243, -243, 0, 729,.. (see A000748). - R. J. Mathar, Feb 25 2023

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Václav Chvátal, A combinatorial theorem in plane geometry, Journal of Combinatorial Theory, Series B 18 (1975), pp. 39-41, doi:10.1016/0095-8956(75)90061-1.
• Clark Kimberling, A Combinatorial Classification of Triangle Centers on the Line at Infinity, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A001477, A002265, A002266, A004526, A008615, A008620, A010761, A010762, A010872, A010873, A010874, A022003, A110532, A110533, A137221 (binomial transform).

Partial sums give A130518.

Cf. A004523 interlaced with A004396.

Apart from the zeros, this is column 3 of A235791.

Programs

Haskell

a002264 n = a002264_list !! n
a002264_list = 0 : 0 : 0 : map (+ 1) a002264_list
-- Reinhard Zumkeller, Nov 06 2012, Apr 16 2012

Magma

[Floor(n/3): n in [0..100]]; // Vincenzo Librandi, Apr 29 2015

Magma

&cat [[n,n,n]: n in [0..30]]; // Bruno Berselli, Apr 29 2015

Maple

seq(i$3,i=0..100); # Robert Israel, Aug 04 2014

Mathematica

Flatten[Table[{n, n, n}, {n, 0, 25}]] (* Harvey P. Dale, Jun 09 2013 *)
Floor[Range[0, 20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Floor[n/3], {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - Cos[2 (n - 2) Pi/3] + Sin[2 (n - 2) Pi/3]/Sqrt[3] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n - 2, -1/2] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 0, 0, 1}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[x^3/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

PARI

a(n)=n\3  /* Jaume Oliver Lafont, Mar 25 2009 */

PARI

v=[0,0];for(n=2,50,v=concat(v,n-2-v[#v]-v[#v-1]));v \\ Derek Orr, Apr 28 2015

Sage

[floor(n/3) for n in range(0,79)] # Zerinvary Lajos, Dec 01 2009

Formula

a(n) = floor(n/3).

a(n) = (3*n-3-sqrt(3)*(1-2*cos(2*Pi*(n-1)/3))*sin(2*Pi*(n-1)/3))/9. - Hieronymus Fischer, Sep 18 2007

a(n) = (n - A010872(n))/3. - Hieronymus Fischer, Sep 18 2007

Complex representation: a(n) = (n - (1 - r^n)*(1 + r^n/(1 - r)))/3 where r = exp(2*Pi/3*i) = (-1 + sqrt(3)*i)/2 and i = sqrt(-1). - Hieronymus Fischer, Sep 18 2007; - corrected by Guenther Schrack, Sep 26 2019

a(n) = Sum_{k=0..n-1} A022003(k). - Hieronymus Fischer, Sep 18 2007

G.f.: x^3/((1-x)*(1-x^3)). - Hieronymus Fischer, Sep 18 2007

a(n) = (n - 1 + 2*sin(4*(n+2)*Pi/3)/sqrt(3))/3. - Jaume Oliver Lafont, Dec 05 2008

For n >= 3, a(n) = floor(log_3(3^a(n-1) + 3^a(n-2) + 3^a(n-3))). - Vladimir Shevelev, Jun 22 2010

a(n) = (n - 3 + A010872(n-1) + A010872(n-2))/3 using Zumkeller's 2008 formula in A010872. - Adriano Caroli, Nov 23 2010

a(n) = A004526(n) - A008615(n). - Reinhard Zumkeller, Apr 28 2014

a(2*n) = A004523(n) and a(2*n+1) = A004396(n). - L. Edson Jeffery, Jul 30 2014

a(n) = n - 2 - a(n-1) - a(n-2) for n > 1 with a(0) = a(1) = 0. - Derek Orr, Apr 28 2015

From Wesley Ivan Hurt, May 27 2015: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4), n > 4.

a(n) = (n - 1 + 0^((-1)^(n/3) - (-1)^n) - 0^((-1)^(n/3)*(-1)^(1/3) + (-1)^n))/3. (End)

a(n) = (3*n - 3 + r^n*(1 - r) + r^(2*n)*(r + 2))/9 where r = (-1 + sqrt(-3))/2. - Guenther Schrack, Sep 26 2019

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