

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
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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(62*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}];


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



