A358997 a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).

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

Offset: 0

Author

Robert Israel, Dec 09 2022

Keywords

• nonn
• look

Comments

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

Examples

			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)).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A012265, A332325.

Programs

Maple

f:= proc(n) local p, k;
  p:= add((-1)^k * x^k/(2*k)!, k=0..n);
  sturm(sturmseq(p,x),x,0,infinity)
end proc:
map(f, [$0..100]);

Mathematica

a[n_] := CountRoots[Sum[(-1)^k*x^k/(2k)!, {k, 0, n}], {x, 0, Infinity}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 12 2023 *)