A332325 Number of Maclaurin polynomials p(2m,x) of cos(x) that have exactly n positive zeros.

3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 4, 5, 4

Offset: 1

Author

Clark Kimberling, Feb 11 2020

Keywords

• nonn

Comments

Maclaurin polynomial p(2m,x) of cos(x) is 1 - x^2/2! + x^4/4! - ... + (-1)^m*x^(2m)/(2m)!.

Examples

			a(1) counts these values of 2m: 2, 6, and 10. The single positive zeros of p(2,x), p(6,x), and p(10,x) are sqrt(2), 1.56990..., and 1.57079..., respectively.
		

Crossrefs

Cf. A332326, A332420, A358997.

Programs

Mathematica

z = 30; p[m_, x_] := Normal[Series[Cos[x], {x, 0, m }]];
t[n_] := x /. NSolve[p[n, x] == 0, x, z];
u[n_] := Select[t[n], Im[#] == 0 && # > 0 &];
v = Table[Length[u[n]], {n, 2, 100, 2}]
Table[Count[v, n], {n, 1, 10}]

Extensions

More terms from Jinyuan Wang, Jan 21 2025