%I #135 Dec 27 2023 11:52:23

%S 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,

%T 11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,

%U 18,19,19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,25

%N Nonnegative integers repeated 3 times.

%C Complement of A010872, since A010872(n) + 3*a(n) = n. - _Hieronymus Fischer_, Jun 01 2007

%C 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

%C 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

%H Michael De Vlieger, <a href="/A002264/b002264.txt">Table of n, a(n) for n = 0..10000</a>

%H Václav Chvátal, <a href="http://dx.doi.org/10.1016/0095-8956(75)90061-1">A combinatorial theorem in plane geometry</a>, Journal of Combinatorial Theory, Series B 18 (1975), pp. 39-41, doi:10.1016/0095-8956(75)90061-1.

%H Clark Kimberling, <a href="https://www.emis.de/journals/JIS/VOL22/Kimberling/kimb9.html">A Combinatorial Classification of Triangle Centers on the Line at Infinity</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.4.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

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

%F 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

%F a(n) = (n - A010872(n))/3. - _Hieronymus Fischer_, Sep 18 2007

%F 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

%F a(n) = Sum_{k=0..n-1} A022003(k). - _Hieronymus Fischer_, Sep 18 2007

%F G.f.: x^3/((1-x)*(1-x^3)). - _Hieronymus Fischer_, Sep 18 2007

%F a(n) = (n - 1 + 2*sin(4*(n+2)*Pi/3)/sqrt(3))/3. - _Jaume Oliver Lafont_, Dec 05 2008

%F 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

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

%F a(n) = A004526(n) - A008615(n). - _Reinhard Zumkeller_, Apr 28 2014

%F a(2*n) = A004523(n) and a(2*n+1) = A004396(n). - _L. Edson Jeffery_, Jul 30 2014

%F a(n) = n - 2 - a(n-1) - a(n-2) for n > 1 with a(0) = a(1) = 0. - _Derek Orr_, Apr 28 2015

%F From _Wesley Ivan Hurt_, May 27 2015: (Start)

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

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

%F 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

%F 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

%p seq(i$3,i=0..100); # _Robert Israel_, Aug 04 2014

%t Flatten[Table[{n, n, n}, {n, 0, 25}]] (* _Harvey P. Dale_, Jun 09 2013 *)

%t Floor[Range[0, 20]/3] (* _Eric W. Weisstein_, Aug 12 2023 *)

%t Table[Floor[n/3], {n, 0, 20}] (* ~~~ *)

%t 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 *)

%t Table[(n - ChebyshevU[n - 2, -1/2] - 1)/3, {n, 0, 20}] (* _Eric W. Weisstein_, Aug 12 2023 *)

%t LinearRecurrence[{1, 0, 1, -1}, {0, 0, 0, 1}, 20] (* _Eric W. Weisstein_, Aug 12 2023 *)

%t CoefficientList[Series[x^3/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 12 2023 *)

%o (PARI) a(n)=n\3 /* _Jaume Oliver Lafont_, Mar 25 2009 */

%o (Sage) [floor(n/3) for n in range(0,79)] # _Zerinvary Lajos_, Dec 01 2009

%o (Haskell)

%o a002264 n = a002264_list !! n

%o a002264_list = 0 : 0 : 0 : map (+ 1) a002264_list

%o -- _Reinhard Zumkeller_, Nov 06 2012, Apr 16 2012

%o (PARI) v=[0,0];for(n=2,50,v=concat(v,n-2-v[#v]-v[#v-1]));v \\ _Derek Orr_, Apr 28 2015

%o (Magma) [Floor(n/3): n in [0..100]]; // _Vincenzo Librandi_, Apr 29 2015

%o (Magma) &cat [[n,n,n]: n in [0..30]]; // _Bruno Berselli_, Apr 29 2015

%Y Cf. A001477, A002265, A002266, A004526, A008615, A008620, A010761, A010762, A010872, A010873, A010874, A022003, A110532, A110533, A137221 (binom. Trans).

%Y Partial sums give A130518.

%Y Cf. A004523 interlaced with A004396.

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

%K nonn,easy

%O 0,7

%A _N. J. A. Sloane_