A193677 Number of negative zeros of minimal polynomials of 2*cos(Pi/n), n>=1.

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

Offset

1,8

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 nonnegative zeros are given by A193676(n).

Links

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

Formula

a(n) = delta(n) - A193676(n), with the degree of C(n,x) given by delta(n)=A055034(n).

These numbers have also been computed employing PIE (principle of inclusion and exclusion) for the three cases mentioned in A193676.

Example

n=1: C(1,x) has only the zero -2, therefore a(1)=1.
n=2: C(2,x) has only a vanishing zero, therefore a(2)=0.
n=5: C(5,x) has the negative zero 2*cos(3*Pi/5) = -2*cos(2*Pi/5)=-(phi-1) with the golden section phi, therefore a(5)=1.
n=8: C(8,x) has the two negative zeros 2*cos(5*Pi/8) =
  -2*cos(3*Pi/8) = -sqrt(2-sqrt(2)) and 2*cos(7*Pi/8) =
  -2*cos(Pi/8)= -sqrt(2+sqrt(2)), therefore a(8)=2.

Crossrefs

Cf. A187360, A055034, A193676.

Sequence in context: A025422 A078640 A006374 * A281855 A137921 A064876

Adjacent sequences: A193674 A193675 A193676 * A193678 A193679 A193680

Keyword

nonn

Author

Wolfdieter Lang, Aug 02 2011

Status

approved