

A193676


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


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
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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 Cpolynomial 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(2sqrt(2)), therefore a(8)=2.


CROSSREFS

Cf. A187360, A055034, A193677.
Sequence in context: A316845 A120481 A219644 * A029291 A022872 A091423
Adjacent sequences: A193673 A193674 A193675 * A193677 A193678 A193679


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Aug 02 2011


STATUS

approved



