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

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, 15, 16, 15, 16, 17, 18, 17, 18, 19, 20, 19, 20, 21, 22, 21, 22, 23, 24, 23, 24, 25, 26, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 31, 32, 31, 32, 33, 34, 33, 34

Offset: 0

Author

Wolfdieter Lang, May 10 2017

Keywords

• nonn
• easy

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A008611, A016627, A049310, A059169, A162330, A183041.

Programs

Mathematica

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

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

Formula

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).

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).

From Colin Barker, May 18 2017: (Start)

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

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

(End)

a(n) = A162330(n-1) for n >= 2. - Michel Marcus, Nov 01 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) (A016627). - Amiram Eldar, Sep 17 2023

a(n) = A183041(n-1) for n>=2. - R. J. Mathar, May 13 2025