A047271 Numbers that are congruent to {0, 3, 5} mod 6.

0, 3, 5, 6, 9, 11, 12, 15, 17, 18, 21, 23, 24, 27, 29, 30, 33, 35, 36, 39, 41, 42, 45, 47, 48, 51, 53, 54, 57, 59, 60, 63, 65, 66, 69, 71, 72, 75, 77, 78, 81, 83, 84, 87, 89, 90, 93, 95, 96, 99, 101, 102, 105, 107, 108, 111, 113, 114, 117, 119, 120, 123, 125

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			G.f. = 3*x^2 + 5*x^3 + 6*x^4 + 9*x^5 + 11*x^6 + 12*x^7 +15*x^8 + 17*x^9 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Programs

Magma

I:=[0, 3, 5, 6]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 26 2012

Maple

A047271:=n->(6*n-4+cos(2*n*Pi/3)-sqrt(3)*sin(2*Pi*n/3))/3: seq(A047271(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016

Mathematica

Select[Range[0,120], MemberQ[{0,3,5}, Mod[#,6]]&] (* Vincenzo Librandi, Apr 26 2012 *)

Formula

From Colin Barker, Mar 13 2012: (Start)

G.f.: x^2*(3+2*x+x^2)/((1-x)^2*(1+x+x^2)).

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

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

a(n) = (6*n-4+cos(2*n*Pi/3)-sqrt(3)*sin(2*Pi*n/3))/3.

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

Sum_{n>=2} (-1)^n/a(n) = log(2)/6 + log(2+sqrt(3))/(2*sqrt(3)) - Pi/12. - Amiram Eldar, Dec 14 2021

Extensions

G.f. corrected by Colin Barker, May 14 2012