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

1,2

Links

Table of n, a(n) for n=1..63.

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

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

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 *)

Prog

(PARI) isok(n) = my(m = n % 6); (m==0) || (m==2) || (m==3); \\ Michel Marcus, Oct 02 2015
(Magma) [n : n in [0..130] | n mod 6 in [0, 2, 3]]; // Vincenzo Librandi, Oct 02 2015

Crossrefs

Cf. A047240, A047242.

Sequence in context: A032711 A135768 A287659 * A111215 A099381 A289943

Adjacent sequences: A047241 A047242 A047243 * A047245 A047246 A047247

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved