A047258 - OEIS

%I #33 Sep 08 2022 08:44:56

%S 0,4,5,6,10,11,12,16,17,18,22,23,24,28,29,30,34,35,36,40,41,42,46,47,

%T 48,52,53,54,58,59,60,64,65,66,70,71,72,76,77,78,82,83,84,88,89,90,94,

%U 95,96,100,101,102,106,107,108,112,113,114,118,119,120,124

%N Numbers that are congruent to {0, 4, 5} 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^2*(4+x+x^2)/((1+x+x^2)*(x-1)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, Apr 13 2015: (Start)

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

%F a(n) = 2n-2 + ((2n-2) mod 3). (End)

%F From _Wesley Ivan Hurt_, Jun 13 2016: (Start)

%F a(n) = 2*n-1-2*sin(2*n*Pi/3)/sqrt(3).

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

%F E.g.f.: 1 + (2*x - 1)*exp(x) - 2*sin(sqrt(3)*x/2)*(cosh(x/2) - sinh(x/2))/sqrt(3). - _Ilya Gutkovskiy_, Jun 14 2016

%F Sum_{n>=2} (-1)^n/a(n) = log(2+sqrt(3))/(2*sqrt(3)) - (3-sqrt(3))*Pi/18. - _Amiram Eldar_, Dec 14 2021

%p A047258:=n->2*n-2+((2*n-2) mod 3): seq(A047258(n), n=1..100); # _Wesley Ivan Hurt_, Apr 13 2015

%t Flatten[#+{0,4,5}&/@(6Range[0,20])] (* _Harvey P. Dale_, Jul 20 2011 *)

%t Select[Range[0, 200], MemberQ[{0, 4, 5}, Mod[#, 6]] &] (* _Vincenzo Librandi_, Apr 14 2015 *)

%o (Magma) [2*n-2+((2*n-2) mod 3) : n in [1..100]]; // _Wesley Ivan Hurt_, Apr 13 2015

%o (PARI) concat (0, Vec(x^2*(4+x+x^2)/((1+x+x^2)*(x-1)^2) + O(x^80))) \\ _Michel Marcus_, Apr 14 2015

%Y Cf. A047245 (complement).

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Wesley Ivan Hurt_, Apr 13 2015