A047244 Numbers that are congruent to {0, 2, 3} mod 6.

0, 2, 3, 6, 8, 9, 12, 14, 15, 18, 20, 21, 24, 26, 27, 30, 32, 33, 36, 38, 39, 42, 44, 45, 48, 50, 51, 54, 56, 57, 60, 62, 63, 66, 68, 69, 72, 74, 75, 78, 80, 81, 84, 86, 87, 90, 92, 93, 96, 98, 99, 102, 104, 105, 108, 110, 111, 114, 116, 117, 120, 122, 123

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. A047240, A047242.

Programs

Magma

[n : n in [0..130] | n mod 6 in [0, 2, 3]]; // Vincenzo Librandi, Oct 02 2015

Maple

A047244:=n->(6*n-7-2*cos(2*n*Pi/3))/3: seq(A047244(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016

Mathematica

Select[Range[0, 200], Mod[#, 6] == 0 || Mod[#, 6] == 2 || Mod[#, 6] == 3 &] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
Select[Range[0, 200], MemberQ[{0, 2, 3}, Mod[#, 6]] &] (* Vincenzo Librandi, Oct 02 2015 *)
LinearRecurrence[{1, 0, 1, -1}, {2, 3, 6, 8}, {0, 20}] (* Eric W. Weisstein, Apr 09 2018 *)
CoefficientList[Series[x (2 + x + 3 x^2)/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 09 2018 *)
Table[(6 n + Cos[2 n Pi/3] + Sqrt[3] Sin[2 n Pi/3] - 1)/3, {n, 0, 20}] (* Eric W. Weisstein, Apr 09 2018 *)

PARI

isok(n) = my(m = n % 6); (m==0) || (m==2) || (m==3); \\ Michel Marcus, Oct 02 2015

Formula

G.f.: x^2*(2+x+3*x^2) / ((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-7-2*cos(2*n*Pi/3))/3.

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

E.g.f.: (9 + (6*x - 7)*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) = (3-sqrt(3))*Pi/18 + log(2+sqrt(3))/(2*sqrt(3)) + log(2)/3. - Amiram Eldar, Dec 14 2021