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A102283
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Period 3: repeat (0,1,-1).
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12
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0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The sequence is the non-principal Dirichlet character of the reduced residue system mod 3. (The other is A011655.) Associated Dirichlet L-functions are L(1,chi)= sum_{n>=1} a(n)/n = A073010, L(2,chi)= sum_{n>=1} a(n)/n^2 = A086724, or L(3,chi)= sum_{n>=1} a(n)/n^3 = A129404. [Jolley eq 310] [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2010]
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REFERENCES
| M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
L. B. W. Jolley, Summation of Series, Dover (1961).
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LINKS
| R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547, Table 2, Table 22 for m=3, r=2.
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FORMULA
| a(n) = A049347(n-1).
a(n)=-a(n-1)-a(n-2); a(0)=0, a(1)=1. G.f.: x/(1+x+x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]
a(n)=-(1/3)*{(n mod 3)-2*[(n+1) mod 3]+[(n+2) mod 3]}, with n>=0. a(n)=-(1/3)*I*sqrt(3)*[ -1/2+(1/2)*I*sqrt(3)]^n+(1/3)*I*sqrt(3)*[ -1/2-(1/2)*I *sqrt(3)]^n, with n>=0 and I=sqrt(-1) [From Paolo P. Lava (paoloplava(AT)gmail.com), Nov 06 2008]
a(n) = -2*sin(4*pi*n/3)/sqrt(3) = 2*sin(8*pi*n/3)/sqrt(3) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
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MAPLE
| ch:=n-> if n mod 3 = 0 then 0; elif n mod 3 = 1 then 1; else -1; fi;
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CROSSREFS
| Sequence in context: A092220 A011655 * A128834 A022928 A000494 A022933
Adjacent sequences: A102280 A102281 A102282 * A102284 A102285 A102286
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 02 2008
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