%I #23 Sep 08 2022 08:44:57

%S 0,1,2,4,5,7,8,9,10,12,13,15,16,17,18,20,21,23,24,25,26,28,29,31,32,

%T 33,34,36,37,39,40,41,42,44,45,47,48,49,50,52,53,55,56,57,58,60,61,63,

%U 64,65,66,68,69,71,72,73,74,76,77,79,80,81,82,84,85,87,88

%N Numbers that are congruent to {0, 1, 2, 4, 5, 7} mod 8.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).

%F G.f.: x^2*(x^5+2*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - _Colin Barker_, Jun 22 2012

%F From _Wesley Ivan Hurt_, Jun 16 2016: (Start)

%F a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.

%F a(n) = (24*n-27+3*cos(n*Pi)+6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18.

%F a(6k) = 8k-1, a(6k-1) = 8k-3, a(6k-2) = 8k-4, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)

%F Sum_{n>=2} (-1)^n/a(n) = (6-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(sqrt(2)+2)/8 - (2-sqrt(2))*Pi/16. - _Amiram Eldar_, Dec 27 2021

%p A047498:=n->(24*n-27+3*cos(n*Pi)+6*cos(n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/18: seq(A047498(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016

%t Select[Range[0,100],MemberQ[{0,1,2,4,5,7},Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,0,0,0,1,-1},{0,1,2,4,5,7,8},100] (* _Harvey P. Dale_, Jul 23 2015 *)

%o (Magma) [n : n in [0..100] | n mod 8 in [0, 1, 2, 4, 5, 7]]; // _Wesley Ivan Hurt_, Jun 16 2016

%Y Cf. A047513, A047581.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_