cp's OEIS Frontend

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).

%I A375054 #4 Oct 02 2024 14:29:03
%S A375054 -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,
%T A375054 26,24,30,28,26,32,30,36,34,32,38,36,34,40,38,44,42,40,46,44,50,48,46,
%U A375054 52,50,48,54,52,58,56,54,60,58,64,62,60,66,64,62,68,66
%N 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).
%C A375054 1
%e A375054 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.
%t A375054 z = 100;
%t A375054 a[n_] := CountRoots[Sum[(-1)^k*x^k/(2 k)!, {k, 0, n}], {x, 0, Infinity}];
%t A375054 t = 2 Table[a[n], {n, 1, z}] ; (* # real zeros of M(n,x) *)
%t A375054 2 Range[z] - t (* # nonreal zeros *)
%t A375054 2 Range[z] - 2 t (* # nonreal zeros minus # real zeros; *)
%Y A375054 Cf. A374987, A376456, A376457, A375053, A375057.
%K A375054 sign
%O A375054 1,1
%A A375054 _Clark Kimberling_, Oct 01 2024