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%I #28 Sep 08 2022 08:44:56
%S 1,4,5,7,10,11,13,16,17,19,22,23,25,28,29,31,34,35,37,40,41,43,46,47,
%T 49,52,53,55,58,59,61,64,65,67,70,71,73,76,77,79,82,83,85,88,89,91,94,
%U 95,97,100,101,103,106,107,109,112,113,115,118,119,121,124
%N Numbers that are congruent to {1, 4, 5} mod 6.
%H Vincenzo Librandi, <a href="/A047259/b047259.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F From _R. J. Mathar_, Feb 21 2009: (Start)
%F G.f.: x*(1+3*x+x^2+x^3)/((1-x)^2*(1+x+x^2)).
%F a(n) = a(n-3) + 6. (End)
%F 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
%F From _Wesley Ivan Hurt_, Jun 11 2016: (Start)
%F a(n) = (6*n-2-cos(2*n*Pi/3)-sqrt(3)*sin(2*n*Pi/3))/3.
%F a(3k) = 6k-1, a(3k-1) = 6k-2, a(3k-2) = 6k-5. (End)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (6-sqrt(3))*Pi/18 + log(2)/6. - _Amiram Eldar_, Dec 16 2021
%p 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
%t 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 *)
%t LinearRecurrence[{1, 0, 1, -1}, {1, 4, 5, 7}, 100] (* _Vincenzo Librandi_, Jun 14 2016 *)
%o (Magma) [n : n in [0..150] | n mod 6 in [1, 4, 5]]; // _Wesley Ivan Hurt_, Jun 11 2016
%Y Cf. A144430 (essentially the same), A010882 (first differences), A080341 (partial sums).
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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