%I #34 Sep 08 2022 08:44:57

%S 1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28,33,34,35,36,41,42,43,44,

%T 49,50,51,52,57,58,59,60,65,66,67,68,73,74,75,76,81,82,83,84,89,90,91,

%U 92,97,98,99,100,105,106,107,108,113,114,115,116,121,122,123

%N Numbers that are congruent to {1, 2, 3, 4} mod 8.

%H Vincenzo Librandi, <a href="/A047454/b047454.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

%F From _Colin Barker_, May 14 2012: (Start)

%F a(n) = (-5-(-1)^n-(1-i)*(-i)^n-(1+i)*i^n+4*n)/2 where i=sqrt(-1).

%F G.f.: x*(1+x+x^2+x^3+4*x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Vincenzo Librandi_, May 15 2012

%F a(2k) = A047463(k), a(2k-1) = A047471(k). - _Wesley Ivan Hurt_, Jun 01 2016

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 + 3*log(2)/8. - _Amiram Eldar_, Dec 23 2021

%p A047454:=n->(-5-I^(2*n)-(1-I)*(-I)^n-(1+I)*I^n+4*n)/2: seq(A047454(n), n=1..100); # _Wesley Ivan Hurt_, Jun 01 2016

%t Select[Range[0,300], MemberQ[{1,2,3,4}, Mod[#,8]]&] (* _Vincenzo Librandi_, May 15 2012 *)

%o (Magma) I:=[1, 2, 3, 4, 9]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // _Vincenzo Librandi_, May 15 2012

%o (PARI) my(x='x+O('x^100)); Vec(x*(1+x+x^2+x^3+4*x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ _Altug Alkan_, Dec 24 2015

%Y Cf. A047463, A047471.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_