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A047403 Numbers that are congruent to {0, 2, 3, 6} mod 8.

%I A047403 #26 Mar 02 2023 12:59:44
%S A047403 0,2,3,6,8,10,11,14,16,18,19,22,24,26,27,30,32,34,35,38,40,42,43,46,
%T A047403 48,50,51,54,56,58,59,62,64,66,67,70,72,74,75,78,80,82,83,86,88,90,91,
%U A047403 94,96,98,99,102,104,106,107,110,112,114,115,118,120,122,123,126,128
%N A047403 Numbers that are congruent to {0, 2, 3, 6} mod 8.
%H A047403 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047403 a(n) = 2*n - ((n mod 4) == 2).
%F A047403 G.f.: x^2*(2+x+3*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 05 2011
%F A047403 From _Wesley Ivan Hurt_, May 24 2016: (Start)
%F A047403 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047403 a(n) = (8*n-9+i^(2*n)+i^(1-n)-i^(1+n))/4, where i=sqrt(-1).
%F A047403 a(2k) = A016825(k-1) for k>0, a(2k-1) = A047470. (End)
%F A047403 E.g.f.: (4 + sin(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - _Ilya Gutkovskiy_, May 25 2016
%F A047403 Sum_{n>=2} (-1)^n/a(n) = (4-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (sqrt(2)-1)*Pi/16. - _Amiram Eldar_, Dec 21 2021
%p A047403 A047403:=n->(8*n-9+I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047403(n), n=1..100); # _Wesley Ivan Hurt_, May 24 2016
%t A047403 Table[(8n-9+I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* _Wesley Ivan Hurt_, May 24 2016 *)
%t A047403 #+{0,2,3,6}&/@(8*Range[0,20])//Flatten (* or *) LinearRecurrence[{1,0,0,1,-1},{0,2,3,6,8},80] (* _Harvey P. Dale_, Mar 02 2023 *)
%o A047403 (Magma) [n : n in [0..150] | n mod 8 in [0, 2, 3, 6]]; // _Wesley Ivan Hurt_, May 24 2016
%Y A047403 Cf. A016825, A047470.
%K A047403 nonn,easy
%O A047403 1,2
%A A047403 _N. J. A. Sloane_