%I #28 Sep 08 2022 08:44:57

%S 0,1,3,5,6,7,8,9,11,13,14,15,16,17,19,21,22,23,24,25,27,29,30,31,32,

%T 33,35,37,38,39,40,41,43,45,46,47,48,49,51,53,54,55,56,57,59,61,62,63,

%U 64,65,67,69,70,71,72,73,75,77,78,79,80,81,83,85,86,87,88

%N Numbers that are congruent to {0, 1, 3, 5, 6, 7} mod 8.

%H Vincenzo Librandi, <a href="/A047585/b047585.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).

%F From _Chai Wah Wu_, Jun 10 2016: (Start)

%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6).

%F G.f.: x^2*(x^4 + x^2 + x + 1)/((x - 1)^2*(x^2 - x + 1)*(x^2 + x + 1 ) ). (End)

%F a(n) = (12*n - 3*sqrt(3)*sin(Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) - 9)/9. - _Ilya Gutkovskiy_, Jun 11 2016

%F a(3k) = 8k-1, a(3k-1) = 8k-2, a(3k-2) = 8k-3, a(3k-3) = 8k-5, a(3k-4) = 8k-7, a(3k-5) = 8k-8. - _Wesley Ivan Hurt_, Jun 16 2016

%F Sum_{n>=2} (-1)^n/a(n) = (3-2*sqrt(2))*Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - _Amiram Eldar_, Dec 27 2021

%p A047585:=n->(12*n - 3*sqrt(3)*sin(Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) - 9)/9: seq(A047585(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016

%t Select[Range[0,100], MemberQ[{0,1,3,5,6,7}, Mod[#,8]]&] (* or *) Complement[Range[0,100], Flatten[Range[{2,4},100,8]]] (* _Harvey P. Dale_, May 01 2012 *)

%t CoefficientList[Series[x (x^4 + x^2 + x + 1) / ((x - 1)^2 (x^2 - x + 1) (x^2 + x + 1)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Jun 18 2016 *)

%o (Magma) [n : n in [0..100] | n mod 8 in [0, 1, 3, 5, 6, 7]]; // _Wesley Ivan Hurt_, Jun 16 2016

%Y Cf. A047517, A047569.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Dec 11 1999