%I #36 Sep 08 2022 08:44:57

%S 0,1,3,4,6,7,8,9,11,12,14,15,16,17,19,20,22,23,24,25,27,28,30,31,32,

%T 33,35,36,38,39,40,41,43,44,46,47,48,49,51,52,54,55,56,57,59,60,62,63,

%U 64,65,67,68,70,71,72,73,75,76,78,79,80,81,83,84,86,87,88,89

%N Numbers that are congruent to {0, 1, 3, 4, 6, 7} mod 8.

%H G. C. Greubel, <a href="/A047517/b047517.txt">Table of n, a(n) for n = 1..1000</a>

%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 From _Chai Wah Wu_, May 30 2016: (Start)

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

%F G.f.: x^2*(x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1). (End)

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

%F a(n) = (24*n-21-3*cos(n*Pi)+2*sqrt(3)*cos((1+4*n)*Pi/6)+6*sin((1-2*n)*Pi/6))/18.

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

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

%p A047424:=n->(24*n-21-3*cos(n*Pi)+2*sqrt(3)*cos((1+4*n)*Pi/6)+6*sin((1-2*n)* Pi/6))/18: seq(A047424(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016

%t LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 4, 6, 7, 8}, 50] (* _G. C. Greubel_, May 30 2016 *)

%t Select[Range[0,200], MemberQ[{0, 1, 3, 4, 6, 7}, Mod[#, 8]] &] (* _Vincenzo Librandi_, May 30 2016 *)

%o (Magma) [n: n in [0..110] | n mod 8 in [0,1,3,4,6,7]]; // _Vincenzo Librandi_, May 30 2016

%o (PARI) my(x='x+O('x^50)); concat([0], Vec(x^2*(x^5 + x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^7 - x^6 - x + 1))) \\ _G. C. Greubel_, Oct 29 2017

%Y Cf. A047428, A047585.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_