A047231 Numbers that are congruent to {0, 3, 4} mod 6.
0, 3, 4, 6, 9, 10, 12, 15, 16, 18, 21, 22, 24, 27, 28, 30, 33, 34, 36, 39, 40, 42, 45, 46, 48, 51, 52, 54, 57, 58, 60, 63, 64, 66, 69, 70, 72, 75, 76, 78, 81, 82, 84, 87, 88, 90, 93, 94, 96, 99, 100, 102, 105, 106, 108, 111, 112, 114, 117, 118, 120, 123, 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).
Crossrefs
Cf. A061347.
Programs
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Magma
[n: n in [0..120] | n mod 6 in [0, 3, 4]]; // Vincenzo Librandi, Jan 06 2013
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Maple
A047231:=n->(6*n-5-cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/3: seq(A047231(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016
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Mathematica
Select[Range[0, 600], MemberQ[{0, 3, 4}, Mod[#, 6]]&] (* Vincenzo Librandi, Jan 06 2013 *) LinearRecurrence[{1,0,1,-1},{0,3,4,6},70] (* Harvey P. Dale, Sep 03 2017 *)
Formula
From R. J. Mathar, Aug 05 2010: (Start)
G.f.: x^2*(3+x+2*x^2) / ( (1+x+x^2)*(x-1)^2 ).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*n+2-(11+A061347(n+1))/3. (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = (6*n-5-cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/3.
a(3k) = 6k-2, a(3k-1) = 6k-3, a(3k-2) = 6k-6. (End)
Sum_{n>=2} (-1)^n/a(n) = log(2)/3 + (1-2/sqrt(3))*Pi/12. - Amiram Eldar, Dec 14 2021