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

0,3

Comments

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

Links

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

Example

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)).

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 *)

Crossrefs

Cf. A012265, A332325.

Sequence in context: A261641 A325622 A060145 * A373461 A257806 A035391

Adjacent sequences: A358994 A358995 A358996 * A358998 A358999 A359000

Keyword

nonn,look

Author

Robert Israel, Dec 09 2022

Status

approved