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

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`%I #14 Nov 12 2023 13:16:40
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`%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,
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`%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,
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`%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
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`%N a(n) is the number of distinct positive real roots of the Maclaurin polynomial of degree 2*n for cos(x).
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`%C It appears that a(n) == n (mod 2) and a(n+2) - a(n) is always either 0 or 2.
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`%H Robert Israel, <a href="/A358997/b358997.txt">Table of n, a(n) for n = 0..250</a>
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`%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)).
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`%p f:= proc(n) local p, k;
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`%p p:= add((-1)^k * x^k/(2*k)!, k=0..n);
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`%p sturm(sturmseq(p,x),x,0,infinity)
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`%p end proc:
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`%p map(f, [$0..100]);
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`%t a[n_] := CountRoots[Sum[(-1)^k*x^k/(2k)!, {k, 0, n}], {x, 0, Infinity}];
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`%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Nov 12 2023 *)
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`%Y Cf. A012265, A332325.
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`%K nonn,look
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`%O 0,3
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`%A _Robert Israel_, Dec 09 2022
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