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

1,2

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).

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

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 *)

Prog

(Magma) [n: n in [0..100] | n mod 6 in [1..4]]; // 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

Crossrefs

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

Complement of A047264. Equals A203016 divided by 3.

Sequence in context: A104576 A332485 A031477 * A317047 A228102 A261412

Adjacent sequences: A047224 A047225 A047226 * A047228 A047229 A047230

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved