A358997 - OEIS

%I #14 Nov 12 2023 13:16:40

%S 0,1,2,1,2,1,2,3,2,3,4,3,4,3,4,5,4,5,6,5,6,5,6,7,6,7,6,7,8,7,8,9,8,9,

%T 8,9,10,9,10,11,10,11,10,11,12,11,12,11,12,13,12,13,14,13,14,13,14,15,

%U 14,15,14,15,16,15,16,17,16,17,16,17,18,17,18,19,18,19,18,19,20,19,20,19,20,21

%N a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).

%C It appears that a(n) == n (mod 2) and a(n+2) - a(n) is always either 0 or 2.

%H Robert Israel, <a href="/A358997/b358997.txt">Table of n, a(n) for n = 0..250</a>

%e a(2) = 2 because the Maclaurin polynomial of degree 4, 1 - x^2/2! + x^4/4!, has two distinct nonnegative real roots, namely sqrt(6-2*sqrt(3)) and sqrt(6+2*sqrt(3)).

%p f:= proc(n) local p, k;

%p   p:= add((-1)^k * x^k/(2*k)!, k=0..n);

%p   sturm(sturmseq(p,x),x,0,infinity)

%p end proc:

%p map(f, [$0..100]);

%t a[n_] := CountRoots[Sum[(-1)^k*x^k/(2k)!, {k, 0, n}], {x, 0, Infinity}];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 12 2023 *)

%Y Cf. A012265, A332325.

%K nonn,look

%O 0,3

%A _Robert Israel_, Dec 09 2022