OFFSET
0,3
COMMENTS
This sequence can be extended periodically to negative values of n.
See a comment on A203571 where a k-family of 2k-periodic sequences P_k has been defined. The present sequence is P_3. - Wolfdieter Lang, Feb 02 2012
LINKS
Rémi Guillaume, Formulas for A193680, and some proofs.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
FORMULA
a(n) = n (mod 3) if (-1)^floor(n/3)=+1 else (3 - n)(mod 3), n>=0. (-1)^floor(n/3) is the parity of the quotient floor(n/3), sometimes denoted by n\3.
O.g.f.: x*(1+2*x+2*x^3+x^4)/(1-x^6).
a(n) = 1-((-1)^(n+1)+cos(Pi*n/3)+3*cos(2*Pi*n/3))/3. - R. J. Mathar, Oct 07 2011, corrected by Vaclav Kotesovec, Feb 19 2023
a(n) = floor((71/364)*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 04 2013
a(n) = floor((4007/333333)*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 2, a(3^e) = 0, and a(p^e) = 1 for p >= 5.
Dirichlet g.f.: zeta(s)*(1+1/2^s-1/3^s-1/6^s). (End)
EXAMPLE
a(8) = 8(mod 3) = 2 because (-1)^floor(8/3)= +1; 8\3 = 2 is even.
a(4) = (3-4)(mod 3) = 2, because (-1)^floor(4/3) is -1; 4\3 = 1 is odd.
MATHEMATICA
Table[{0, 1, 2, 0, 2, 1}, {15}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
PadRight[{}, 120, {0, 1, 2, 0, 2, 1}] (* Harvey P. Dale, Jul 25 2020 *)
PROG
(PARI) a(n)=[0, 1, 2, 0, 2, 1][n%6+1] \\ Charles R Greathouse IV, Oct 07 2011
CROSSREFS
KEYWORD
nonn,easy,mult,changed
AUTHOR
Wolfdieter Lang, Sep 30 2011
EXTENSIONS
Keyword:mult added by Andrew Howroyd, Jul 31 2018
STATUS
approved