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A120325 Period 6: repeat [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, 0, 0, 1, 0, 1, 0, 0, 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). - 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..85.

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.

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

FORMULA

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). - 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

a(n) = (n+3)^2 mod (5+(-1)^n)/2. - Wesley Ivan Hurt, Oct 31 2014

a(n) = sin(n*Pi/3)^2*(2-4*cos(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 19 2016

E.g.f.: 2*(cosh(x) - cos(sqrt(3)*x/2)*cosh(x/2))/3. - Ilya Gutkovskiy, Jun 20 2016

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) ; # R. J. Mathar, Jul 31 2010

MATHEMATICA

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 *)

PROG

(MAGMA) [(n+3)^2 mod (2+((n+1) mod 2)) : n in [0..100]]; // Wesley Ivan Hurt, Oct 31 2014

CROSSREFS

Cf. A100044.

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 December 7 17:16 EST 2016. Contains 278890 sequences.