This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A286717 #19 Sep 08 2022 08:46:19
%S A286717 0,1,2,3,2,3,4,5,6,5,6,7,8,9,8,9,10,11,12,11,12,13,14,15,14,15,16,17,
%T A286717 18,17,18,19,20,21,20,21,22,23,24,23,24,25,26,27,26,27,28,29,30,29,30,
%U A286717 31,32,33,32,33,34,35,36,35,36,37,38,39,38,39,40,41,42
%N A286717 a(n) is the number of zeros of the Chebyshev S(n, x) polynomial (A049310) in the open interval (-phi, +phi), with the golden section phi = (1 + sqrt(5))/2.
%C A286717 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 A286717 G. C. Greubel, <a href="/A286717/b286717.txt">Table of n, a(n) for n = 0..5000</a>
%H A286717 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A286717 a(n) = 2*b(n) if n is even and 1 + 2*b(n) if n is odd with b(n) = floor(n/2) - floor((n+1)/6) = A286716(n). See the g.f. for {b(n)}_{n>=0} there.
%F A286717 From _Colin Barker_, May 18 2017: (Start)
%F A286717 G.f.: x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
%F A286717 a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.
%F A286717 (End)
%e A286717 a(4) = 2: S(4, x) = 1+x^4-3*x^2, and only two of the four zeros -phi, -1/phi, +1/phi, phi are in the open interval (-phi, +phi), the other two are at the borders.
%t A286717 CoefficientList[Series[x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)), {x, 0, 50}], x] (* _G. C. Greubel_, Mar 08 2018 *)
%t A286717 LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,3,2,3},80] (* _Harvey P. Dale_, Aug 20 2020 *)
%o A286717 (PARI) concat(0, Vec(x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^100))) \\ _Colin Barker_, May 18 2017
%o A286717 (Magma) m:=80; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+x^2-x^3+x^4)/((1-x)^2*(1+x+x^2+x^3+x^4)))); // _G. C. Greubel_, Mar 08 2018
%Y A286717 Cf. A008611(n-1) (1), A285869 (sqrt(2)), A285872 (sqrt(3)).
%K A286717 nonn,easy
%O A286717 0,3
%A A286717 _Wolfdieter Lang_, May 13 2017