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A358997
a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).
1
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
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
Sequence in context: A261641 A325622 A060145 * A373461 A257806 A035391
KEYWORD
nonn,look
AUTHOR
Robert Israel, Dec 09 2022
STATUS
approved