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A047582 Numbers that are congruent to {3, 5, 6, 7} mod 8.

%I A047582 #16 Sep 08 2022 08:44:57
%S A047582 3,5,6,7,11,13,14,15,19,21,22,23,27,29,30,31,35,37,38,39,43,45,46,47,
%T A047582 51,53,54,55,59,61,62,63,67,69,70,71,75,77,78,79,83,85,86,87,91,93,94,
%U A047582 95,99,101,102,103,107,109,110,111,115,117,118,119,123,125
%N A047582 Numbers that are congruent to {3, 5, 6, 7} mod 8.
%H A047582 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047582 From _Wesley Ivan Hurt_, May 29 2016: (Start)
%F A047582 G.f.: x*(3+2*x+x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
%F A047582 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047582 a(n) = (8*n+1-i^(2*n)-(2-i)*i^(-n)-(2+i)*i^n)/4 where i=sqrt(-1).
%F A047582 a(2k) = A047550(k), a(2k-1) = A047398(k). (End)
%F A047582 E.g.f.: (2 + sin(x) - 2*cos(x) + sinh(x) + 4*x*exp(x))/2. - _Ilya Gutkovskiy_, May 30 2016
%F A047582 Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 - log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - _Amiram Eldar_, Dec 26 2021
%p A047582 A047582:=n->(8*n+1-I^(2*n)-(2-I)*I^(-n)-(2+I)*I^n)/4: seq(A047582(n), n=1..100); # _Wesley Ivan Hurt_, May 29 2016
%t A047582 Table[(8n+1-I^(2n)-(2-I)*I^(-n)-(2+I)*I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 29 2016 *)
%o A047582 (Magma) [n : n in [0..150] | n mod 8 in [3, 5, 6, 7]]; // _Wesley Ivan Hurt_, May 29 2016
%Y A047582 Cf. A047398, A047550.
%K A047582 nonn,easy
%O A047582 1,1
%A A047582 _N. J. A. Sloane_