A087509 Number of k such that (k*n) == 2 (mod 3) for 0 <= k <= n.

0, 0, 1, 0, 1, 2, 0, 2, 3, 0, 3, 4, 0, 4, 5, 0, 5, 6, 0, 6, 7, 0, 7, 8, 0, 8, 9, 0, 9, 10, 0, 10, 11, 0, 11, 12, 0, 12, 13, 0, 13, 14, 0, 14, 15, 0, 15, 16, 0, 16, 17, 0, 17, 18, 0, 18, 19, 0, 19, 20, 0, 20, 21, 0, 21, 22, 0, 22, 23, 0, 23, 24, 0, 24, 25, 0, 25, 26, 0, 26, 27, 0, 27, 28, 0, 28

Offset: 0

Author

Paul Barry, Sep 11 2003

Keywords

• easy
• nonn

Examples

			a(8) = #{1,4,7} = 3.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Crossrefs

Cf. A087507, A087508.

Programs

Mathematica

{#-1,1+#,0}[[Mod[#,3,1]]]/3&/@Range[0, 99] (* Federico Provvedi, Jun 15 2021 *)
LinearRecurrence[{0,0,2,0,0,-1},{0,0,1,0,1,2},100] (* Harvey P. Dale, May 04 2023 *)

PARI

a(n) = sum(k=0, n, (k*n % 3)==2); \\ Michel Marcus, Sep 25 2017

Formula

a(n) = Sum_{k=0..n} [(k*n) == 2 (mod 3)];

a(n) = n - 2*(floor(n/3) + 1)*(1 - cos(2*Pi*n/3))/3 - floor(n/3)*(5 + 4*cos(2*Pi*n/3))/3.

a(n) = n - A087507(n) - A087508(n).

G.f.: x^2*(x^2+1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Mar 31 2013

a(n) = 2*sin(n*Pi/3)*(sqrt(3)*cos(n*Pi) + 2*n*sin(n*Pi/3))/9. - Wesley Ivan Hurt, Sep 24 2017