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A285869 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial in the open interval (-sqrt(2), +sqrt(2)).

%I A285869 #43 May 13 2025 11:57:37
%S A285869 0,1,2,1,2,3,4,3,4,5,6,5,6,7,8,7,8,9,10,9,10,11,12,11,12,13,14,13,14,
%T A285869 15,16,15,16,17,18,17,18,19,20,19,20,21,22,21,22,23,24,23,24,25,26,25,
%U A285869 26,27,28,27,28,29,30,29,30,31,32,31,32,33,34,33,34
%N A285869 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial in the open interval (-sqrt(2), +sqrt(2)).
%C A285869 See a May 06 2017 comment on A049310 where these problems are considered which originated in a conjecture by _Michel Lagneau_ (see A008611) on Fibonacci polynomials.
%H A285869 G. C. Greubel, <a href="/A285869/b285869.txt">Table of n, a(n) for n = 0..1000</a>
%H A285869 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A285869 a(n) = 2*b(n) if n is even, else a(n) = 1 + 2*b(n), with b(n) = floor(n/2) - floor((n + 1)/4) = A059169(n+1).
%F A285869 G.f. for {b(n)}: Sum_{n>=0} b(n)*x^n = x^2*(1 - x + x^2)/((1 - x)*(1 - x^4)) (see A059169).
%F A285869 From _Colin Barker_, May 18 2017: (Start)
%F A285869 G.f.: x*(1 + x - x^2 + x^3) / ((1 - x)^2*(1 + x)*(1 + x^2)).
%F A285869 a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.
%F A285869 (End)
%F A285869 a(n) = A162330(n-1) for n >= 2. - _Michel Marcus_, Nov 01 2017
%F A285869 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) (A016627). - _Amiram Eldar_, Sep 17 2023
%F A285869 a(n) = A183041(n-1) for n>=2. - _R. J. Mathar_, May 13 2025
%t A285869 Table[2 (Floor[n/2] - Floor[(n + 1)/4]) + Boole[OddQ@ n], {n, 0, 52}] (* _Michael De Vlieger_, May 10 2017 *)
%o A285869 (PARI) concat(0, Vec(x*(1 + x - x^2 + x^3) / ((1 - x)^2*(1 + x)*(1 + x^2)) + O(x^100))) \\ _Colin Barker_, May 18 2017
%Y A285869 Cf. A008611, A016627, A049310, A059169, A162330, A183041.
%K A285869 nonn,easy
%O A285869 0,3
%A A285869 _Wolfdieter Lang_, May 10 2017