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A002264
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Integers repeated 3 times.
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74
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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; internal format)
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OFFSET
| 0,7
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COMMENTS
| Complement of A010872, since A010872(n)+3*a(n)=n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| floor(n/3), n>=0.
a(n) = -1 + sum{k=0..n} {1/9*[ -2*(k mod 3)+[(k+1) mod 3]+4*[(k+2) mod 3]]} - Paolo P. Lava (paoloplava(AT)gmail.com), Jun 20 2007
a(n)=(3n-3-sqrt(3)*(1-2cos(2*pi*(n-1)/3))*sin(2*pi*(n-1)/3)))/9. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 18 2007
a(n)=(n-A010872(n))/3. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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 (Hieronymus.Fischer(AT)gmx.de), Sep 18 2007
a(n)=sum{0<=k<n, A022003(k)}. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Sep 18 2007
G.f.: g(x)=x^3/((1-x)(1-x^3)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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 (azarian(AT)evansville.edu), Nov 08 2007
a(n)=(n-1+2sin(4(n+2)pi/3)/sqrt(3))/3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
For n>=3, a(n)=floor(log_3(3^a(n-1)+3^a(n-2)+3^a(n-3))) [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 22 2010]
a(n)=[n-3+A010872(n-1)+A010872(n-2)]/3 using Zumkeller 2008 formula in A010872. -- Adriano Caroli (adriano_caroli(AT)virgilio.it), Nov 23 2010
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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 (paoloplava(AT)gmail.com), Jun 20 2007
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PROG
| (PARI) a(n)=n\3 [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 25 2009]
(Other) sage: [floor(n/3) - 1 for n in xrange(3, 79)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2009]
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CROSSREFS
| Cf. A008620.
Cf. A004526, A002265, A002266, A010761, A010762, A110532, A110533.
Partial sums: A130518. Other related sequences: A004526, A010872, A010873, A010874.
Sequence in context: A079001 A032615 A086161 * A008620 A104581 A113675
Adjacent sequences: A002261 A002262 A002263 * A002265 A002266 A002267
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Clarified my formulas Mohammad K. Azarian (azarian(AT)evansville.edu), Aug 01 2009
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