A010875 a(n) = n mod 6.

0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0

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 2) + 2(floor(n/2) mod 3) = A000035(n) +2*A010872(A004526(n));

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

A010875:=n->n mod 6; seq(A010875(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2014

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

Partial sums: A130484. Other related sequences A130481, A130482, A130483, A130485.

Cf. also A079979, A097325, A122841.

Sequence in context: A369183 A030567 A049265 * A260187 A257687 A309957

Adjacent sequences: A010872 A010873 A010874 * A010876 A010877 A010878

Keyword

nonn,easy

Author

N. J. A. Sloane

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

approved