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A047519 Numbers that are congruent to {1, 2, 3, 4, 6, 7} mod 8.

%I A047519 #22 Sep 08 2022 08:44:57
%S A047519 1,2,3,4,6,7,9,10,11,12,14,15,17,18,19,20,22,23,25,26,27,28,30,31,33,
%T A047519 34,35,36,38,39,41,42,43,44,46,47,49,50,51,52,54,55,57,58,59,60,62,63,
%U A047519 65,66,67,68,70,71,73,74,75,76,78,79,81,82,83,84,86,87
%N A047519 Numbers that are congruent to {1, 2, 3, 4, 6, 7} mod 8.
%H A047519 G. C. Greubel, <a href="/A047519/b047519.txt">Table of n, a(n) for n = 1..1000</a>
%H A047519 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,1,-1).
%F A047519 From _Chai Wah Wu_, May 30 2016: (Start)
%F A047519 a(n) = a(n-1) + a(n-6) - a(n-7), for n > 7.
%F A047519 G.f.: x*(x^6 + x^5 + 2*x^4 + x^3 + x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)
%F A047519 From _Wesley Ivan Hurt_, Jun 16 2016: (Start)
%F A047519 a(n) = (24*n-15-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n)*Pi/6))/18.
%F A047519 a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-4, a(6k-3) = 8k-5, a(6k-4) = 8k-6, a(6k-5) = 8k-7. (End)
%F A047519 Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-1)*Pi/16 + log(2)/4 + sqrt(2)*log(3-2*sqrt(2))/16. - _Amiram Eldar_, Dec 28 2021
%p A047519 A047519:=n->(24*n-15-3*cos(n*Pi)-2*sqrt(3)*cos((1-4*n)*Pi/6)+6*sin((1+2*n)
%p A047519 *Pi/6))/18: seq(A047519(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016
%t A047519 LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 7, 9}, 50] (* _G. C. Greubel_, May 30 2016 *)
%o A047519 (Magma) [n : n in [0..100] | n mod 8 in [1, 2, 3, 4, 6, 7]]; // _Wesley Ivan Hurt_, Jun 16 2016
%Y A047519 Cf. A047422, A047504.
%K A047519 nonn,easy
%O A047519 1,2
%A A047519 _N. J. A. Sloane_