A047245 Numbers that are congruent to {1, 2, 3} mod 6.

1, 2, 3, 7, 8, 9, 13, 14, 15, 19, 20, 21, 25, 26, 27, 31, 32, 33, 37, 38, 39, 43, 44, 45, 49, 50, 51, 55, 56, 57, 61, 62, 63, 67, 68, 69, 73, 74, 75, 79, 80, 81, 85, 86, 87, 91, 92, 93, 97, 98, 99, 103, 104, 105, 109, 110, 111, 115, 116, 117, 121, 122, 123

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(k)^m is a term iff {a(k) is odd and m is a nonnegative integer} or {m is in A004273}. - Jerzy R Borysowicz, May 08 2023

Links

• David Lovler, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A047240, A047244, A047258 (complement).

Programs

Magma

[2*n-3+((2*n-3) mod 3) : n in [1..100]]; // Wesley Ivan Hurt, Apr 13 2015

Maple

A047245:=n->2*n-3+((2*n-3) mod 3): seq(A047245(n), n=1..100); # Wesley Ivan Hurt, Apr 13 2015

Mathematica

Select[Range[0, 200], Mod[#, 6] == 1 || Mod[#, 6] == 2 || Mod[#, 6] == 3 &] (* Vladimir Joseph Stephan Orlovsky, Jul 07 2011 *)
Flatten[Table[{6n + 1, 6n + 2, 6n + 3}, {n, 0, 19}]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[0, 200], MemberQ[{1, 2, 3}, Mod[#, 6]] &] (* Vincenzo Librandi, Apr 14 2015 *)

PARI

a(n) = 3*floor((n-1)/3) + n; \\ David Lovler, Aug 03 2022

Formula

From Johannes W. Meijer, Jun 07 2011: (Start)

a(n) = ceiling(n/3) + ceiling((n-1)/3) + ceiling((n-2)/3) + 3*ceiling((n-3)/3).

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

a(n) = 3*floor((n-1)/3) + n. - Gary Detlefs, Dec 22 2011

From Wesley Ivan Hurt, Apr 13 2015: (Start)

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

a(n) = 2*n-3 + ((2*n-3) mod 3). (End)

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

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

a(3k) = 6k-3, a(3k-1) = 6k-4, a(3k-2) = 6k-5. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (9-2*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) - log(2)/6. - Amiram Eldar, Dec 14 2021