OFFSET
0,3
COMMENTS
Period 6: repeat [0, 1, 2, 3, 4, 5].
The rightmost digit in the base-6 representation of n. - Hieronymus Fischer, Jun 11 2007
[a(n) * a(m)] mod 6 == a(n*m mod 6) == a(n*m). - Jon Perry, Nov 11 2014
If n > 3 and (a(n) is in {0,2,3,4}), then n is not prime. - Jean-Marc Rebert, Jul 22 2015, corrected by M. F. Hasler, Jul 24 2015
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..65538
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).
FORMULA
Complex representation: a(n) = (1/6) * (1 - r^n) * Sum_{k = 1..6} k * Product_{1 <= m < 6, m <> k} (1-r^(n-m)), where r = exp((Pi/3)*i) = (1 + sqrt(3)*i)/2 and i = sqrt(-1).
Trigonometric representation: a(n) = (16/3)^2 * (sin(n*Pi/6))^2 * Sum_{k = 1..6} k * Product_{1 <= m < 6, m<>k} (sin((n-m)*Pi/6))^2.
G.f.: g(x) = (Sum_{k = 1..6} k*x^k)/(1-x^6).
Also: g(x) = x*(5*x^6 - 6*x^5 + 1)/((1 - x^6)*(1 - x)^2). - Hieronymus Fischer, May 31 2007
a(n) = (n mod 3) + 3(floor(n/3) mod 2) = A010872(n) +3*A000035(A002264(n)). - Hieronymus Fischer, Jun 11 2007
a(n) = 2.5 - 0.5*(-1)^n - cos(Pi*n/3) - 3^0.5*sin(Pi*n/3) -cos(2*Pi*n/3) - 3^0.5/3*sin(2*Pi*n/3). - Richard Choulet, Dec 11 2008
a(n) = n^3 mod 6. - Zerinvary Lajos, Oct 29 2009
a(n) = floor(12345/999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013
a(n) = floor(373/9331*6^(n+1)) mod 6. - Hieronymus Fischer, Jan 04 2013
a(n) = 5/2 - (-1)^n/2 - 2*0^((-1)^(n/6 - 1/12 + (-1)^n/12) - (-1)^(n/2 - 1/4 +(-1)^n/4)) + 2*0^((-1)^(n/6 + 1/4 + (-1)^n/12) + (-1)^(n/2 - 1/4 + (-1)^n/4)). - Wesley Ivan Hurt, Jun 23 2015
E.g.f.: -sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 2*cosh(x/2)*cos(sqrt(3)*x/2). - Robert Israel, Jul 22 2015
MAPLE
MATHEMATICA
Mod[Range[0, 100], 6] (* Wesley Ivan Hurt, Jul 06 2014 *)
PROG
(Sage) [power_mod(n, 3, 6 )for n in range(0, 81)] # Zerinvary Lajos, Oct 29 2009
(PARI) a(n)=n%6 \\ Charles R Greathouse IV, Dec 05 2011
(MAGMA) [n mod 6: n in [0..100]]; // Wesley Ivan Hurt, Jul 06 2014
(Scheme) (define (A010875 n) (modulo n 6)) ;; Antti Karttunen, Dec 22 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Formulas 1 to 6 re-edited for better readability by Hieronymus Fischer, Dec 05 2011
More terms from Antti Karttunen, Dec 22 2017
STATUS
editing