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A120325 Periodic sequence 0, 0, 1, 0, 1, 0. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Dirichlet series for the principal character mod 6: L(s,chi) = sum_{n=1..infinity} 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). [From R. J. Mathar, Jul 31 2010]

REFERENCES

L. B. W. Jolley, Summation of Series, Dover (1961).

LINKS

Table of n, a(n) for n=0..77.

FORMULA

a(n) = (1/3)*(sin(n*Pi/6)+sin(7*n*Pi/6))^2, with n>=0.

G.f.: x^2(1+x^2)/((1+x)(1-x)(1+x+x^2)(1-x+x^2)). a(n+6)=a(n). [From R. J. Mathar, Nov 22 2008]

a(n) = (n+3)*Fibonacci(n+3) mod 2. - Gary Detlefs, Dec 13 2010

a(n) = 0 if n mod 6 = 0, else a(n) = n mod 2 +(-1)^n. - Gary Detlefs, Dec 13 2010

EXAMPLE

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)*((3^.5)/2+(3^.5)/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)*((3^.5)/2+(3^.5)/2)^2 = 1.

a(5)= (1/3)*(sin(5*Pi/6)+sin(35*Pi/6)^2 = (1/3)*(1/2-1/2)^2 = 0.

MAPLE

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) ; [From R. J. Mathar, Jul 31 2010]

CROSSREFS

Sequence in context: A059125 A111406 A156731 * A144598 A144606 A060510

Adjacent sequences:  A120322 A120323 A120324 * A120326 A120327 A120328

KEYWORD

easy,nonn

AUTHOR

Paolo P. Lava and Giorgio Balzarotti, Jun 21 2006

STATUS

approved

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Last modified April 17 20:01 EDT 2014. Contains 240655 sequences.