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A083219 a(n) = n - 2*floor(n/4).

%I A083219 #58 Aug 27 2025 08:17:19
%S A083219 0,1,2,3,2,3,4,5,4,5,6,7,6,7,8,9,8,9,10,11,10,11,12,13,12,13,14,15,14,
%T A083219 15,16,17,16,17,18,19,18,19,20,21,20,21,22,23,22,23,24,25,24,25,26,27,
%U A083219 26,27,28,29,28,29,30,31,30,31,32,33,32,33,34,35,34,35,36,37,36,37,38
%N A083219 a(n) = n - 2*floor(n/4).
%C A083219 Conjecture: number of roots of P(x) = x^n - x^(n-1) - x^(n-2) - ... - x - 1 in the left half-plane. - _Michel Lagneau_, Apr 09 2013
%C A083219 a(n) is n+2 with its second least significant bit removed (see A021913(n+2) for that bit). - _Kevin Ryde_, Dec 13 2019
%H A083219 Altug Alkan, <a href="/A083219/b083219.txt">Table of n, a(n) for n = 0..10000</a>
%H A083219 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A083219 a(n) = A083220(n)/2.
%F A083219 a(n) = a(n-1) + n mod 2 + (n mod 4 - 1)*(1 - n mod 2), a(0) = 0.
%F A083219 G.f.: x*(1+x+x^2-x^3)/((1-x)^2*(1+x)*(1+x^2)). - _R. J. Mathar_, Aug 28 2008
%F A083219 a(n) = n - A129756(n). - _Michel Lagneau_, Apr 09 2013
%F A083219 Bisection: a(2*k) = 2*floor((n+2)/4), a(2*k+1) = a(2*k) + 1, k >= 0. - _Wolfdieter Lang_, May 08 2017
%F A083219 a(n) = (2*n + 3 - 2*cos(n*Pi/2) - cos(n*Pi) - 2*sin(n*Pi/2))/4. - _Wesley Ivan Hurt_, Oct 02 2017
%F A083219 a(n) = A162330(n+2) - 1 = A285869(n+3) - 1. - _Kevin Ryde_, Dec 13 2019
%F A083219 E.g.f.: ((1 + x)*cosh(x) - cos(x) + (2 + x)*sinh(x) - sin(x))/2. - _Stefano Spezia_, May 27 2021
%F A083219 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 1. - _Amiram Eldar_, Aug 21 2023
%p A083219 a:= n-> n - 2*floor(n/4):
%p A083219 seq(a(n), n=0..74);  # _Alois P. Heinz_, Jan 24 2021
%t A083219 Array[# - 2 Floor[#/4] &, 75, 0] (* _Michael De Vlieger_, Oct 02 2017 *)
%t A083219 LinearRecurrence[{1,0,0,1,-1},{0,1,2,3,2},80] (* _Harvey P. Dale_, Oct 01 2018 *)
%o A083219 (PARI) a(n) = n - n\4*2 \\ _Charles R Greathouse IV_, Jun 11 2015
%Y A083219 Cf. A083220, A129756, A162751 (second highest bit removed).
%Y A083219 Cf. A021913, A162330, A285869.
%Y A083219 Essentially the same as A018837.
%K A083219 nonn,easy,changed
%O A083219 0,3
%A A083219 _Reinhard Zumkeller_, Apr 22 2003