%I #41 Sep 17 2023 10:00:26

%S 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 15,16,15,16,17,18,17,18,19,20,19,20,21,22,21,22,23,24,23,24,25,26,25,

%U 26,27,28,27,28,29,30,29,30,31,32,31,32,33,34,33,34

%N a(n) is the number of zeros of the Chebyshev S(n, x) polynomial in the open interval (-sqrt(2), +sqrt(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="/A285869/b285869.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

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

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

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>4.

%F (End)

%F a(n) = A162330(n-1) for n >= 2. - _Michel Marcus_, Nov 01 2017

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) (A016627). - _Amiram Eldar_, Sep 17 2023

%t Table[2 (Floor[n/2] - Floor[(n + 1)/4]) + Boole[OddQ@ n], {n, 0, 52}] (* _Michael De Vlieger_, May 10 2017 *)

%o (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 Cf. A008611, A016627, A049310, A059169, A162330, A183041.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, May 10 2017