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

1,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A335001 A368659 A153085 * A287658 A039577 A370861

Adjacent sequences: A047256 A047257 A047258 * A047260 A047261 A047262

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved