A375054 Let M(n,x) denote the Maclaurin polynomial of degree 2n for cos x. Let u(n) be the number of nonreal zeros of M(n,x) and v(n) the number of real zeros of M(n,x). Then a(n) = u(n) - v(n).

-2, -4, 2, 0, 6, 4, 2, 8, 6, 4, 10, 8, 14, 12, 10, 16, 14, 12, 18, 16, 22, 20, 18, 24, 22, 28, 26, 24, 30, 28, 26, 32, 30, 36, 34, 32, 38, 36, 34, 40, 38, 44, 42, 40, 46, 44, 50, 48, 46, 52, 50, 48, 54, 52, 58, 56, 54, 60, 58, 64, 62, 60, 66, 64, 62, 68, 66

Offset

1,1

Comments

1

Links

Table of n, a(n) for n=1..67.

Example

The 6 zeros of the Maclaurin polynomial x^2/2! - x^4/4! - x^6/6! are approximately {-3.92 - 1.28 i, -3.92 + 1.2 i, -1.56, 1.56, 3.92 - 1.28 i, 3.92 + 1.28 i}; there are 4 nonreal zero and 2 real zeros, so that a(3) = 4 - 2 = 2.

Mathematica

z = 100;
a[n_] := CountRoots[Sum[(-1)^k*x^k/(2 k)!, {k, 0, n}], {x, 0, Infinity}];
t = 2 Table[a[n], {n, 1, z}] ; (* # real zeros of M(n, x) *)
2 Range[z] - t (* # nonreal zeros *)
2 Range[z] - 2 t (* # nonreal zeros minus # real zeros; *)

Crossrefs

Cf. A374987, A376456, A376457, A375053, A375057.

Sequence in context: A356165 A366589 A335764 * A202069 A364529 A300329

Adjacent sequences: A375051 A375052 A375053 * A375055 A375056 A375057

Keyword

sign

Author

Clark Kimberling, Oct 01 2024

Status

approved