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
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).
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