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Numbers that are congruent to {1, 2, 4} mod 6.
4

%I #17 Sep 08 2022 08:44:56

%S 1,2,4,7,8,10,13,14,16,19,20,22,25,26,28,31,32,34,37,38,40,43,44,46,

%T 49,50,52,55,56,58,61,62,64,67,68,70,73,74,76,79,80,82,85,86,88,91,92,

%U 94,97,98,100,103,104,106,109,110,112,115,116,118,121,122,124

%N Numbers that are congruent to {1, 2, 4} mod 6.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F G.f.: x*(1+x)*(2*x^2+1)/((1+x+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, Jun 10 2016: (Start)

%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.

%F a(n) = (6*n-5-cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3.

%F a(3k) = 6k-2, a(3k-1) = 6k-4, a(3k-2) = 6k-5. (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/6 + log(2+sqrt(3))/(2*sqrt(3)) - log(2)/3. - _Amiram Eldar_, Dec 14 2021

%p A047236:=n->(6*n-5-cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3: seq(A047236(n), n=1..100); # _Wesley Ivan Hurt_, Jun 10 2016

%t Select[Range[0, 150], MemberQ[{1, 2, 4}, Mod[#, 6]] &] (* _Wesley Ivan Hurt_, Jun 10 2016 *)

%o (Magma) [n : n in [0..150] | n mod 6 in [1, 2, 4]]; // _Wesley Ivan Hurt_, Jun 10 2016

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_