A047259 Numbers that are congruent to {1, 4, 5} mod 6.

1, 4, 5, 7, 10, 11, 13, 16, 17, 19, 22, 23, 25, 28, 29, 31, 34, 35, 37, 40, 41, 43, 46, 47, 49, 52, 53, 55, 58, 59, 61, 64, 65, 67, 70, 71, 73, 76, 77, 79, 82, 83, 85, 88, 89, 91, 94, 95, 97, 100, 101, 103, 106, 107, 109, 112, 113, 115, 118, 119, 121, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A144430 (essentially the same), A010882 (first differences), A080341 (partial sums).

Programs

Magma

[n : n in [0..150] | n mod 6 in [1, 4, 5]]; // Wesley Ivan Hurt, Jun 11 2016

Maple

A047259:=n->(6*n-2-cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/3: seq(A047259(n), n=1..100); # Wesley Ivan Hurt, Jun 11 2016

Mathematica

Select[Range[200], MemberQ[{1,4,5}, Mod[#,6]]&] (* or *) LinearRecurrence[ {1,0,1,-1}, {1,4,5,7}, 100] (* Harvey P. Dale, Feb 16 2015 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 4, 5, 7}, 100] (* Vincenzo Librandi, Jun 14 2016 *)

Formula

From R. J. Mathar, Feb 21 2009: (Start)

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

a(n) = a(n-3) + 6. (End)

a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=1, a(2)=4, a(3)=5, a(4)=7. - Harvey P. Dale, Feb 16 2015

From Wesley Ivan Hurt, Jun 11 2016: (Start)

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

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (6-sqrt(3))*Pi/18 + log(2)/6. - Amiram Eldar, Dec 16 2021