A047490 - OEIS

%I #28 Jan 31 2024 17:54:38

%S 0,1,2,3,5,7,8,9,10,11,13,15,16,17,18,19,21,23,24,25,26,27,29,31,32,

%T 33,34,35,37,39,40,41,42,43,45,47,48,49,50,51,53,55,56,57,58,59,61,63,

%U 64,65,66,67,69,71,72,73,74,75,77,79,80,81,82,83,85,87,88

%N Numbers that are congruent to {0, 1, 2, 3, 5, 7} mod 8.

%H Michael De Vlieger, <a href="/A047490/b047490.txt">Table of n, a(n) for n = 1..10001</a>

%H Giulio Cerbai, <a href="https://arxiv.org/abs/2401.10027">Pattern-avoiding modified ascent sequences</a>, arXiv:2401.10027 [math.CO], 2024. See p. 28.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).

%F G.f.: x^2*(x^4+x^3+x^2+1)/((x-1)^2*(x^2-x+1)*(x^2+x+1)). - _Colin Barker_, Jun 22 2012

%F From _Wesley Ivan Hurt_, Jun 16 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) = (24*n-30+6*sqrt(3)*cos((1-2n)*Pi/6)+2*sqrt(3)*cos((1+4n)*Pi/6))/18.

%F a(6k) = 8k-1, a(6k-1) = 8k-3, 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) = (2*sqrt(2)-3)*Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - _Amiram Eldar_, Dec 26 2021

%p A047490:=n->(24*n-30+6*sqrt(3)*cos((1-2*n)*Pi/6)+2*sqrt(3)*cos((1+4*n)*Pi/6))/18: seq(A047490(n), n=1..100); # _Wesley Ivan Hurt_, Jun 16 2016

%t Select[Range[0, 100], MemberQ[{0, 1, 2, 3, 5, 7}, Mod[#, 8]] &] (* _Wesley Ivan Hurt_, Jun 16 2016 *)

%o (Magma) [n : n in [0..100] | n mod 8 in [0, 1, 2, 3, 5, 7]]; // _Wesley Ivan Hurt_, Jun 16 2016

%Y Cf. A047450, A047549.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_