|
%I #26 Sep 08 2022 08:44:57
%S 0,1,2,3,4,6,8,9,10,11,12,14,16,17,18,19,20,22,24,25,26,27,28,30,32,
%T 33,34,35,36,38,40,41,42,43,44,46,48,49,50,51,52,54,56,57,58,59,60,62,
%U 64,65,66,67,68,70,72,73,74,75,76,78,80,81,82,83,84,86,88
%N Numbers that are congruent to {0, 1, 2, 3, 4, 6} mod 8.
%C The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4, etc. - _Bruno Berselli_, Nov 28 2012
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).
%F a(n+1) = 6*floor(n/3)+(n mod 3). - _Gary Detlefs_, Mar 09 2010
%F G.f.: x^2*(1+x^2+2*x^4) / ( (1+x+x^2)*(x^2-x+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 05 2011
%F From _Wesley Ivan Hurt_, Jun 15 2016: (Start)
%F a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6) for n>6.
%F a(n) = (12*n-18+sqrt(3)*(3*sin(n*Pi/3)+sin(2*n*Pi/3)))/9.
%F a(6k) = 8k-2, a(6k-1) = 8k-4, a(6k-2) = 8k-5, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
%F Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/8 + 3*log(2)/4. - _Amiram Eldar_, Dec 26 2021
%p A047420:=n->(12*n-18+sqrt(3)*(3*sin(n*Pi/3)+sin(2*n*Pi/3)))/9: seq(A047420(n), n=1..100); # _Wesley Ivan Hurt_, Jun 15 2016
%t Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 4, 6}, Mod[#, 8]] &] (* _Wesley Ivan Hurt_, Jun 15 2016 *)
%t LinearRecurrence[{2,-2,2,-2,2,-1},{0,1,2,3,4,6},70] (* _Harvey P. Dale_, Aug 11 2021 *)
%o (Magma) [n : n in [0..100] | n mod 8 in [0..4] cat [6]]; // _Wesley Ivan Hurt_, Jun 15 2016
%Y Cf. A047549, A047602.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
|
