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%I #34 Sep 08 2022 08:44:57
%S 0,1,2,4,6,7,8,9,10,12,14,15,16,17,18,20,22,23,24,25,26,28,30,31,32,
%T 33,34,36,38,39,40,41,42,44,46,47,48,49,50,52,54,55,56,57,58,60,62,63,
%U 64,65,66,68,70,71,72,73,74,76,78,79,80,81,82,84,86,87,88
%N Numbers that are congruent to {0, 1, 2, 4, 6, 7} 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
%C Nonnegative m such that floor(k*(m/4)^2) = k*floor((m/4)^2), where k = 2 or 3. - _Bruno Berselli_, Dec 03 2015
%H G. C. Greubel, <a href="/A047513/b047513.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).
%F From _Chai Wah Wu_, May 29 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), n>6.
%F G.f.: x^2*(x^2 + 1)^2/((x - 1)^2*(x^2 - x + 1)*(x^2 + x + 1)). (End)
%F From _Wesley Ivan Hurt_, Jun 15 2016: (Start)
%F a(n) = (24*n-24-2*sqrt(3)*(cos((1-4*n)*Pi/6)-3*cos((2*n+1)*Pi/6)))/18.
%F a(6k) = 8k-1, a(6k-1) = 8k-2, a(6k-2) = 8k-4, a(6k-3) = 8k-6, a(6k-4) = 8k-7, a(6k-5) = 8k-8. (End)
%F Sum_{n>=2} (-1)^n/a(n) = (4-sqrt(2))*log(2)/8 + sqrt(2)*log(sqrt(2)+2)/4. - _Amiram Eldar_, Dec 27 2021
%p A047513:=n->(24*n-24-2*sqrt(3)*(cos((1-4*n)*Pi/6)-3*cos((2*n+1)*Pi/6)))/18: seq(A047513(n), n=1..100); # _Wesley Ivan Hurt_, Jun 15 2016
%t Select[Range[0, 75], Function[k, Mod[#, 8] == k] /@ Nor[3, 5] &] (* or *) Select[Range[0, 75], Function[k, Floor[k (#/4)^2] == k Floor[(#/4)^2]] /@ Or[2, 3] &] (* _Michael De Vlieger_, Dec 03 2015 *)
%t Select[Range[0,100], MemberQ[{0,1,2,4,6,7}, Mod[#,8]]&] (* _Harvey P. Dale_, Apr 26 2016 *)
%t LinearRecurrence[{2, -2, 2, -2, 2, -1}, {0, 1, 2, 4, 6, 7}, 50] (* _G. C. Greubel_, May 29 2016 *)
%o (Magma) [n : n in [0..100] | n mod 8 in [0, 1, 2, 4, 6, 7]]; // _Wesley Ivan Hurt_, May 29 2016
%Y Cf. A047498, A047581.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_