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A120325
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Period 6: repeat [0, 0, 1, 0, 1, 0].
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10
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0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0
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OFFSET
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0,1
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COMMENTS
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Dirichlet series for the principal character mod 6: L(s,chi) = Sum_{n>=1} a(n+3)/n^s = (1 + 1/6^s - 1/2^s - 1/3^s) Riemann-zeta(s), e.g., L(2,chi) = A100044, L(4,chi) = 5*Pi^4/486, L(6,chi) = 91*Pi^6/87480. See Jolley eq (313) and arXiv:1008.2547 L(m=6,r=1,s). - R. J. Mathar, Jul 31 2010
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REFERENCES
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L. B. W. Jolley, Summation of Series, Dover (1961).
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LINKS
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FORMULA
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a(n) = (1/3)*(sin(n*Pi/6) + sin(7*n*Pi/6))^2.
G.f.: x^2*(1+x^2)/((1+x)*(1-x)*(1+x+x^2)*(1-x+x^2)).
a(n+6) = a(n). (End)
a(n) = ((n+3)*Fibonacci(n+3)) mod 2. - Gary Detlefs, Dec 13 2010
a(n) = 0 if n mod 6 = 0, otherwise a(n) = n mod 2 + (-1)^n. - Gary Detlefs, Dec 13 2010
E.g.f.: 2*(cosh(x) - cos(sqrt(3)*x/2)*cosh(x/2))/3. - Ilya Gutkovskiy, Jun 20 2016
a(n) = [A276086(n) == 3 (mod 6)], where [ ] is the Iverson bracket.
(End)
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EXAMPLE
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a(0) = (1/3)*(sin(0) + sin(0))^2 = 0.
a(1) = (1/3)*(sin(Pi/6) + sin(7*Pi/6))^2 = (1/3)*(1/2 - 1/2)^2 = 0.
a(2) = (1/3)*(sin(Pi/3) + sin(7*Pi/3))^2 = (1/3)*((sqrt(3))/2 + (sqrt(3))/2)^2 = 1.
a(3) = (1/3)*(sin(Pi/2) + sin(7*Pi/2))^2 = (1/3)*(1 - 1)^2 = 0.
a(4) = (1/3)*(sin(2*Pi/3) + sin(14*Pi/3))^2 = (1/3)*((sqrt(3))/2 + (sqrt(3))/2)^2 = 1.
a(5) = (1/3)*(sin(5*Pi/6) + sin(35*Pi/6)^2 = (1/3)*(1/2 - 1/2)^2 = 0.
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MAPLE
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P:=proc(n)local i, j; for i from 0 by 1 to n do j:=1/3*(sin(i*Pi/6)+sin(7*i*Pi/6))^2; print(j); od; end: P(20);
seq(abs(numtheory[jacobi](n, 6)), n=3..150) ; # R. J. Mathar, Jul 31 2010
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MATHEMATICA
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Table[Mod[(n + 3)^2, (5 + (-1)^n)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Oct 31 2014 *)
PadRight[{}, 120, {0, 0, 1, 0, 1, 0}] (* Harvey P. Dale, Oct 05 2016 *)
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PROG
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(Magma) [(n+3)^2 mod (2+((n+1) mod 2)) : n in [0..100]]; // Wesley Ivan Hurt, Oct 31 2014
(Python)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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