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A047482 Numbers that are congruent to {1, 2, 5, 7} mod 8.

%I A047482 #34 Sep 08 2022 08:44:57
%S A047482 1,2,5,7,9,10,13,15,17,18,21,23,25,26,29,31,33,34,37,39,41,42,45,47,
%T A047482 49,50,53,55,57,58,61,63,65,66,69,71,73,74,77,79,81,82,85,87,89,90,93,
%U A047482 95,97,98,101,103,105,106,109,111,113,114,117,119,121,122,125
%N A047482 Numbers that are congruent to {1, 2, 5, 7} mod 8.
%H A047482 Vincenzo Librandi, <a href="/A047482/b047482.txt">Table of n, a(n) for n = 1..1000</a>
%H A047482 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A047482 From _Colin Barker_, May 14 2012: (Start)
%F A047482 a(n) = (-5 - (-1)^n + (-i)^n + i^n + 8*n)/4 where i=sqrt(-1).
%F A047482 G.f.: x*(1 + x + 3*x^2 + 2*x^3 + x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)
%F A047482 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Vincenzo Librandi_, May 17 2012
%F A047482 a(2k) = A047524(k), a(2k-1) = A016813(k-1) for k>0. - _Wesley Ivan Hurt_, Jun 01 2016
%F A047482 E.g.f.: (2 + cos(x) + (4*x - 2)*sinh(x) + (4*x - 3)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 01 2016
%F A047482 Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - _Amiram Eldar_, Dec 24 2021
%p A047482 A047482:=n->(-5-(-1)^n+(-I)^n+I^n+8*n)/4: seq(A047482(n), n=1..100); # _Wesley Ivan Hurt_, Jun 01 2016
%t A047482 Select[Range[0,300], MemberQ[{1,2,5,7}, Mod[#,8]]&] (* _Vincenzo Librandi_, May 17 2012 *)
%t A047482 LinearRecurrence[{1,0,0,1,-1},{1, 2, 5, 7, 9},100] (* _G. C. Greubel_, Jun 01 2016 *)
%o A047482 (Magma) I:=[1, 2, 5, 7, 9]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // _Vincenzo Librandi_, May 17 2012
%o A047482 (PARI) my(x='x+O('x^100)); Vec(x*(1+x+3*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ _Altug Alkan_, Dec 24 2015
%Y A047482 Cf. A016813, A047524.
%K A047482 nonn,easy
%O A047482 1,2
%A A047482 _N. J. A. Sloane_