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Numbers that are congruent to {0, 1, 2, 5, 6, 7} mod 8.
3

%I #25 Sep 08 2022 08:44:57

%S 0,1,2,5,6,7,8,9,10,13,14,15,16,17,18,21,22,23,24,25,26,29,30,31,32,

%T 33,34,37,38,39,40,41,42,45,46,47,48,49,50,53,54,55,56,57,58,61,62,63,

%U 64,65,66,69,70,71,72,73,74,77,78,79,80,81,82,85,86,87,88

%N Numbers that are congruent to {0, 1, 2, 5, 6, 7} mod 8.

%H G. C. Greubel, <a href="/A047581/b047581.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 + x^3 + 3*x^2 + x + 1)/(x^7 - x^6 - x + 1). (End)

%F a(n) = (8*n + (-1)^n - 2*sqrt(3)*sin(Pi*n/3) - 4*sin(2*Pi*(n+1)/3)/sqrt(3) + 2*cos(Pi*n/3) - 7)/6. - _Ilya Gutkovskiy_, May 30 2016

%F a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-3, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. - _Wesley Ivan Hurt_, Jun 16 2016

%F Sum_{n>=2} (-1)^n/a(n) = (12-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)-1)*Pi/16. - _Amiram Eldar_, Dec 27 2021

%p A047581:=n->(8*n+(-1)^n-2*sqrt(3)*sin(Pi*n/3)-4*sin(2*Pi*(n+1)/3)/sqrt(3)

%p +2*cos(Pi*n/3)-7)/6: seq(A047581(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016

%t LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 5, 6, 7, 8}, 50] (* _G. C. Greubel_, May 30 2016 *)

%o (Magma) [n : n in [0..100] | n mod 8 in [0, 1, 2, 5, 6, 7]]; // _Wesley Ivan Hurt_, Jun 16 2016

%Y Cf. A047498, A047513.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_