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%I #34 Sep 08 2022 08:44:56
%S 0,1,4,6,7,10,12,13,16,18,19,22,24,25,28,30,31,34,36,37,40,42,43,46,
%T 48,49,52,54,55,58,60,61,64,66,67,70,72,73,76,78,79,82,84,85,88,90,91,
%U 94,96,97,100,102,103,106,108,109,112,114,115,118,120,121,124
%N Numbers that are congruent to {0, 1, 4} mod 6.
%H Vincenzo Librandi, <a href="/A047234/b047234.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 Equals partial sums of (0, 1, 3, 2, 1, 3, 2, 1, 3, 2, ...). - _Gary W. Adamson_, Jun 19 2008
%F G.f.: x^2*(1+x)*(2*x+1)/((1+x+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, Jun 14 2016: (Start)
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
%F a(n) = (6*n-7+cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3.
%F a(3k) = 6k-2, a(3k-1) = 6k-5, a(3k-2) = 6k-6. (End)
%F Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(3))*Pi/18 + log(2)/3 + log(2+sqrt(3))/(2*sqrt(3)). - _Amiram Eldar_, Dec 14 2021
%p A047234:=n->(6*n-7+cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3: seq(A047234(n), n=1..100); # _Wesley Ivan Hurt_, Jun 14 2016
%t Select[Range[0, 200], Mod[#, 6] == 0 || Mod[#, 6] == 1 || Mod[#, 6] == 4 &] (* _Vladimir Joseph Stephan Orlovsky_, Jul 07 2011 *)
%t LinearRecurrence[{1, 0, 1, -1}, {0, 1, 4, 6}, 100] (* _Vincenzo Librandi_, Jun 15 2016 *)
%t #+{0,1,4}&/@(6*Range[0,20])//Flatten (* _Harvey P. Dale_, Jul 25 2019 *)
%o (PARI) a(n)=(n-1)\3*6+[4,0,1][n%3+1] \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) [n : n in [0..150] | n mod 6 in [0, 1, 4]]; // _Wesley Ivan Hurt_, Jun 14 2016
%Y Cf. A047240, A047242.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
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