A193676 Number of nonnegative zeros of minimal polynomials of 2*cos(Pi/n), n>=1.

0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 3, 3, 2, 4, 4, 3, 5, 4, 2, 5, 6, 4, 5, 6, 5, 6, 7, 4, 8, 8, 4, 8, 6, 6, 9, 9, 6, 8, 10, 6, 11, 10, 6, 11, 12, 8, 10, 10, 8, 12, 13, 9, 10, 12, 8, 14, 15, 8, 15, 15, 10, 16, 12, 10, 17, 16, 10, 12

Offset

1,7

Comments

The coefficient array for the minimal polynomials of 2*cos(Pi/n), n>=1, called C(n,x), is given in A187360. The zeros are also given there.

C(2,x)=x is the only C-polynomial with a vanishing zero.

The number of negative zeros is given by A193677(n) = delta(n)-a(n), with the degree of C(n,x) given by delta(n)=A055034(n).

Links

Table of n, a(n) for n=1..70.

Formula

a(n) is the number of nonnegative zeros of C(n,x), n>=1.

Computation, employing PIE (principle of inclusion and exclusion), for the three cases: n even, n odd, congruent 1 (mod 4), and n odd, congruent 3 (mod 4).

Example

m=1: C(1,x) has only a negative zero -2, therefore a(1)=0.
n=2: C(2,x) has only a vanishing zero, therefore a(2)=1.
n=5: C(5,x) has one positive zero, namely 2*cos(Pi/5), the golden section, therefore a(5)=1.
n=8: C(8,x) has two positive zeros: 2*cos(Pi/8) = sqrt(2+sqrt(2)) and 2*cos(3*Pi/8)=sqrt(2-sqrt(2)), therefore a(8)=2.

Crossrefs

Cf. A187360, A055034, A193677.

Sequence in context: A369067 A356647 A219644 * A029291 A333529 A369363

Adjacent sequences: A193673 A193674 A193675 * A193677 A193678 A193679

Keyword

nonn

Author

Wolfdieter Lang, Aug 02 2011

Status

approved