A047227 Numbers that are congruent to {1, 2, 3, 4} mod 6.

1, 2, 3, 4, 7, 8, 9, 10, 13, 14, 15, 16, 19, 20, 21, 22, 25, 26, 27, 28, 31, 32, 33, 34, 37, 38, 39, 40, 43, 44, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69, 70, 73, 74, 75, 76, 79, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 97, 98

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(k)^m is a term for k and m in N. - Jerzy R Borysowicz, Apr 18 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A008621, A047235, A047241, A047245, A047246, A047247, A093148.

Complement of A047264. Equals A203016 divided by 3.

Programs

Magma

[n: n in [0..100] | n mod 6 in [1..4]]; // Vincenzo Librandi, Jan 06 2013

Maple

A047227:=n->(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A047227(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Mathematica

Complement[Range[100], Flatten[Table[{6n - 1, 6n}, {n, 0, 15}]]] (* Alonso del Arte, Jul 07 2011 *)
Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,1,0,0,1]^(n-1)*[1;2;3;4;7])[1,1] \\ Charles R Greathouse IV, May 03 2023

Formula

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

a(n) = floor((n+2)/4) + floor((n+1)/4) + floor(n/4) + 2*floor((n-1)/4) + floor((n+3)/4).

G.f.: x*(1 + x + x^2 + x^3 + 2*x^4)/(x^5 - x^4 - x + 1). (End)

From Wesley Ivan Hurt, May 20 2016: (Start)

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

a(n) = (6n - 5 - i^(2n) + (1+i)*i^(1-n) + (1-i)*i^(n-1))/4 where i=sqrt(-1).

a(2n) = A047235(n), a(2n-1) = A047241(n). (End)

E.g.f.: (4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016

From Wesley Ivan Hurt, May 21 2016: (Start)

a(n) = A047246(n) + 1.

a(n+2) - a(n+1) = A093148(n) for n>0.

a(1-n) = - A047247(n). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/12 + 2*log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021