A286717 - OEIS

%I #19 Sep 08 2022 08:46:19

%S 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 18,17,18,19,20,21,20,21,22,23,24,23,24,25,26,27,26,27,28,29,30,29,30,

%U 31,32,33,32,33,34,35,36,35,36,37,38,39,38,39,40,41,42

%N 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 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 G. C. Greubel, <a href="/A286717/b286717.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).

%F 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 From _Colin Barker_, May 18 2017: (Start)

%F G.f.: x*(1 + x + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = a(n-1) + a(n-5) - a(n-6) for n>5.

%F (End)

%e 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 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 LinearRecurrence[{1,0,0,0,1,-1},{0,1,2,3,2,3},80] (* _Harvey P. Dale_, Aug 20 2020 *)

%o (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 (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 Cf. A008611(n-1) (1), A285869 (sqrt(2)), A285872 (sqrt(3)).

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, May 13 2017