OFFSET
0,1
COMMENTS
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
REFERENCES
L. B. W. Jolley, Summation of Series, Dover (1961).
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..100000
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, 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.
From R. J. Mathar, Nov 22 2008: (Start)
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
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
a(n) = sign((n-3) mod 2) * sign((n-3) mod 3). - Wesley Ivan Hurt, Feb 04 2022
From Antti Karttunen, Dec 03 2022: (Start)
a(n) = 1 - A093719(n).
a(n) = [A276086(n) == 3 (mod 6)], where [ ] is the Iverson bracket.
(End)
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)*((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.
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
(Python)
def A120325(n): return int(not (n+3) % 6 & 3 ^ 1) # Chai Wah Wu, May 25 2022
(PARI) A120325(n) = ((n%3)&&!(n%2)); \\ Antti Karttunen, Dec 03 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paolo P. Lava and Giorgio Balzarotti, Jun 21 2006
EXTENSIONS
Data section extended up to a(120) by Antti Karttunen, Dec 03 2022
STATUS
approved