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

Author

Clark Kimberling, Oct 01 2024

Keywords

• sign

Comments

1

Examples

			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.
		

Crossrefs

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

Programs

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; *)