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A002264 Integers repeated 3 times. 89
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

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

LINKS

Table of n, a(n) for n=0..75.

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.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

FORMULA

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

a(n) = -1 + Sum{k=0..n} (-2*(k mod 3)+((k+1) mod 3)+4*((k+2) mod 3))/9. - Paolo P. Lava, Jun 20 2007

a(n) = (3n-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)=(3n-(1-r^n)*(1+r^n/(1-r)))/9 where r=exp(2*Pi/3*i)=(-1+sqrt(3)*i)/2 and i=sqrt(-1). - Hieronymus Fischer, Sep 18 2007

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

Also, floor((n^3-1)/(3*n^2)) (n>=1) will produce this sequence. Moreover, floor((n^3-n^2)/(3*n^2-2*n)) (n>=1) will produce this sequence as well. - Mohammad K. Azarian, Nov 08 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)

MAPLE

P:=proc(n) local a, i, k; for i from 0 by 1 to n do a:=-1+sum('1/9*(-2*(k mod 3)+((k+1) mod 3)+4*((k+2) mod 3))', 'k'=0..i); print(a); od; end: P(100); # Paolo P. Lava, Jun 20 2007

# Alternative:

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 *)

PROG

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

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

(Haskell)

a002264 n = a002264_list !! n

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

-- Reinhard Zumkeller, Nov 06 2012, Apr 16 2012

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

(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

CROSSREFS

Cf. A008620, A004526, A002265, A002266, A010761, A010762, A110532, A110533, A010872, A010873, A010874.

Partial sums: A130518.

A004523 interlaced with A004396.

Sequence in context: A032615 A261231 A086161 * A008620 A104581 A261916

Adjacent sequences:  A002261 A002262 A002263 * A002265 A002266 A002267

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Clarified my formulas Mohammad K. Azarian, Aug 01 2009

STATUS

approved

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Last modified May 22 13:18 EDT 2017. Contains 286872 sequences.