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A002264 Nonnegative integers repeated 3 times.

%I A002264 #146 Mar 28 2025 17:31:53
%S A002264 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 A002264 11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,
%U A002264 18,19,19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,25
%N A002264 Nonnegative integers repeated 3 times.
%C A002264 Complement of A010872, since A010872(n) + 3*a(n) = n. - _Hieronymus Fischer_, Jun 01 2007
%C A002264 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 A002264 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 A002264 Michael De Vlieger, <a href="/A002264/b002264.txt">Table of n, a(n) for n = 0..10000</a>
%H A002264 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 A002264 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 A002264 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A002264 a(n) = floor(n/3).
%F A002264 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 A002264 a(n) = (n - A010872(n))/3. - _Hieronymus Fischer_, Sep 18 2007
%F A002264 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 A002264 a(n) = Sum_{k=0..n-1} A022003(k). - _Hieronymus Fischer_, Sep 18 2007
%F A002264 G.f.: x^3/((1-x)*(1-x^3)). - _Hieronymus Fischer_, Sep 18 2007
%F A002264 a(n) = (n - 1 + 2*sin(4*(n+2)*Pi/3)/sqrt(3))/3. - _Jaume Oliver Lafont_, Dec 05 2008
%F A002264 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 A002264 a(n) = (n - 3 + A010872(n-1) + A010872(n-2))/3 using Zumkeller's 2008 formula in A010872. - _Adriano Caroli_, Nov 23 2010
%F A002264 a(n) = A004526(n) - A008615(n). - _Reinhard Zumkeller_, Apr 28 2014
%F A002264 a(2*n) = A004523(n) and a(2*n+1) = A004396(n). - _L. Edson Jeffery_, Jul 30 2014
%F A002264 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 A002264 From _Wesley Ivan Hurt_, May 27 2015: (Start)
%F A002264 a(n) = a(n-1) + a(n-3) - a(n-4), n > 4.
%F A002264 a(n) = (n - 1 + 0^((-1)^(n/3) - (-1)^n) - 0^((-1)^(n/3)*(-1)^(1/3) + (-1)^n))/3. (End)
%F A002264 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 A002264 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 A002264 seq(i$3,i=0..100); # _Robert Israel_, Aug 04 2014
%t A002264 Flatten[Table[{n, n, n}, {n, 0, 25}]] (* _Harvey P. Dale_, Jun 09 2013 *)
%t A002264 Floor[Range[0, 20]/3] (* _Eric W. Weisstein_, Aug 12 2023 *)
%t A002264 Table[Floor[n/3], {n, 0, 20}] (* _Eric W. Weisstein_, Aug 12 2023 *)
%t A002264 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 A002264 Table[(n - ChebyshevU[n - 2, -1/2] - 1)/3, {n, 0, 20}] (* _Eric W. Weisstein_, Aug 12 2023 *)
%t A002264 LinearRecurrence[{1, 0, 1, -1}, {0, 0, 0, 1}, 20] (* _Eric W. Weisstein_, Aug 12 2023 *)
%t A002264 CoefficientList[Series[x^3/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 12 2023 *)
%o A002264 (PARI) a(n)=n\3  /* _Jaume Oliver Lafont_, Mar 25 2009 */
%o A002264 (Sage) [floor(n/3) for n in range(0,79)] # _Zerinvary Lajos_, Dec 01 2009
%o A002264 (Haskell)
%o A002264 a002264 n = a002264_list !! n
%o A002264 a002264_list = 0 : 0 : 0 : map (+ 1) a002264_list
%o A002264 -- _Reinhard Zumkeller_, Nov 06 2012, Apr 16 2012
%o A002264 (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 A002264 (Magma) [Floor(n/3): n in [0..100]]; // _Vincenzo Librandi_, Apr 29 2015
%o A002264 (Magma) &cat [[n,n,n]: n in [0..30]]; // _Bruno Berselli_, Apr 29 2015
%Y A002264 Cf. A001477, A002265, A002266, A004526, A008615, A008620, A010761, A010762, A010872, A010873, A010874, A022003, A110532, A110533, A137221 (binomial transform).
%Y A002264 Partial sums give A130518.
%Y A002264 Cf. A004523 interlaced with A004396.
%Y A002264 Apart from the zeros, this is column 3 of A235791.
%K A002264 nonn,easy
%O A002264 0,7
%A A002264 _N. J. A. Sloane_