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A332325 Number of Maclaurin polynomials p(2m,x) of cos x that have exactly n positive zeros. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A255176 A288177 A064042 * A194882 A096343 A069812

Adjacent sequences:  A332322 A332323 A332324 * A332326 A332327 A332328

KEYWORD

nonn,hard,more

AUTHOR

Clark Kimberling, Feb 11 2020

STATUS

approved

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Last modified July 12 11:25 EDT 2020. Contains 335658 sequences. (Running on oeis4.)