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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..40.

Example

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 1.41421..., 1.56990..., and 1.57079..., respectively.

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}]

Crossrefs

Cf. A332326, A332420.

Sequence in context: A288177 A349992 A064042 * A194882 A096343 A351911

Adjacent sequences: A332322 A332323 A332324 * A332326 A332327 A332328

Keyword

nonn,hard,more

Author

Clark Kimberling, Feb 11 2020

Status

approved