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A047533 Numbers that are congruent to {1, 2, 3, 7} mod 8.

%I A047533 #19 Sep 08 2022 08:44:57
%S A047533 1,2,3,7,9,10,11,15,17,18,19,23,25,26,27,31,33,34,35,39,41,42,43,47,
%T A047533 49,50,51,55,57,58,59,63,65,66,67,71,73,74,75,79,81,82,83,87,89,90,91,
%U A047533 95,97,98,99,103,105,106,107,111,113,114,115,119,121,122,123
%N A047533 Numbers that are congruent to {1, 2, 3, 7} mod 8.
%H A047533 G. C. Greubel, <a href="/A047533/b047533.txt">Table of n, a(n) for n = 1..1000</a>
%H A047533 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047533 From _Wesley Ivan Hurt_, May 29 2016: (Start)
%F A047533 G.f.: x*(1+x+x^2+4*x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
%F A047533 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A047533 a(n) = (8*n-7+i^(2*n)+(1+2*i)*i^(-n)+(1-2*i)*i^n)/4 where i=sqrt(-1).
%F A047533 a(2k) = A047524(k), a(2k-1) = A047471(k). (End)
%F A047533 E.g.f.: (2 + 2*sin(x) + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - _Ilya Gutkovskiy_, May 29 2016
%F A047533 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - _Amiram Eldar_, Dec 23 2021
%p A047533 A047533:=n->(8*n-7+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047533(n), n=1..100); # _Wesley Ivan Hurt_, May 29 2016
%t A047533 Table[(8n-7+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* _Wesley Ivan Hurt_, May 29 2016 *)
%t A047533 LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 7, 9}, 50] (* _G. C. Greubel_, May 29 2016 *)
%o A047533 (Magma) [n : n in [0..150] | n mod 8 in [1, 2, 3, 7]]; // _Wesley Ivan Hurt_, May 29 2016
%Y A047533 Cf. A047471, A047524.
%K A047533 nonn,easy
%O A047533 1,2
%A A047533 _N. J. A. Sloane_