A047267 Numbers that are congruent to {0, 2, 5} mod 6.

0, 2, 5, 6, 8, 11, 12, 14, 17, 18, 20, 23, 24, 26, 29, 30, 32, 35, 36, 38, 41, 42, 44, 47, 48, 50, 53, 54, 56, 59, 60, 62, 65, 66, 68, 71, 72, 74, 77, 78, 80, 83, 84, 86, 89, 90, 92, 95, 96, 98, 101, 102, 104, 107, 108, 110, 113, 114, 116, 119, 120, 122, 125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A011655. [Gary Detlefs, Mar 19 2010]

Programs

Magma

[n : n in [0..150] | n mod 6 in [0, 2, 5]]; // Wesley Ivan Hurt, Jun 13 2016

Maple

seq(3*n-3*floor(n/3)-(n^2 mod 3), n=0..54); # Gary Detlefs, Mar 19 2010

Mathematica

Select[Range[0,110], MemberQ[{0,2,5}, Mod[#,6]]&] (* or *) LinearRecurrence[{1,0,1,-1}, {0,2,5,6}, 60] (* Harvey P. Dale, Aug 31 2015 *)

Formula

a(n) = 3*n-3*floor(n/3)-(n^2 mod 3), with offset 0. - Gary Detlefs, Mar 19 2010

G.f.: x^2*(x+2)*(1+x) / ((1+x+x^2)*(x-1)^2). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jun 13 2016: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

a(n) = (6*n-5+2*cos(2*n*Pi/3))/3.

a(3k) = 6k-1, a(3k-1) = 6k-4, a(3k-2) = 6k-6. (End)

E.g.f.: (3 + (6*x - 5)*exp(x) + 2*cos(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2)))/3. - Ilya Gutkovskiy, Jun 14 2016

Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + log(2+sqrt(3))/(2*sqrt(3)) - (3-sqrt(3))*Pi/18. - Amiram Eldar, Dec 14 2021