login
Numbers that are congruent to {0, 2, 3, 4} mod 8.
2

%I #31 Sep 08 2022 08:44:57

%S 0,2,3,4,8,10,11,12,16,18,19,20,24,26,27,28,32,34,35,36,40,42,43,44,

%T 48,50,51,52,56,58,59,60,64,66,67,68,72,74,75,76,80,82,83,84,88,90,91,

%U 92,96,98,99,100,104,106,107,108,112,114,115,116,120,122,123

%N Numbers that are congruent to {0, 2, 3, 4} mod 8.

%H Vincenzo Librandi, <a href="/A047456/b047456.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 G.f.: x^2*(2+x+x^2+4*x^3)/((1-x)^2*(1+x)*(1+x^2)). - _Colin Barker_, May 13 2012

%F a(n) = (-11-(-1)^n-(2-i)*(-i)^n-(2+i)*i^n+8*n)/4 where i=sqrt(-1). - _Colin Barker_, May 14 2012

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - _Vincenzo Librandi_, May 16 2012

%F a(2k) = A047463(k), a(2k-1) = A047470(k). - _Wesley Ivan Hurt_, May 31 2016

%F E.g.f.: (8 + sin(x) - 2*cos(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - _Ilya Gutkovskiy_, May 31 2016

%F Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - _Amiram Eldar_, Dec 21 2021

%p A047456:=n->(-11-(-1)^n-(2-I)*(-I)^n-(2+I)*I^n+8*n)/4: seq(A047456(n), n=1..100); # _Wesley Ivan Hurt_, May 31 2016

%t Select[Range[0,300], MemberQ[{0,2,3,4}, Mod[#,8]]&] (* _Vincenzo Librandi_, May 16 2012 *)

%o (Magma) I:=[0, 2, 3, 4, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // _Vincenzo Librandi_, May 16 2012

%Y Cf. A047463, A047470.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_