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A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1. 3
0, 1, -2, 1, -3, 0, 1, -2, 0, 1, 5, 0, -5, 0, 1, -1, 1, -7, 0, 14, 0, -7, 0, 1, 2, 0, -4, 0, 1, -3, 0, 9, 0, -6, 0, 1, -1, 1, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 1, 0, -4, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 1, -2, -1, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, 2, 0, -16, 0, 20, 0, -8, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1, 1, -3, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

s(n) := 2*sin(Pi/n) is, for n >= 2, the length ratio side/R of the regular n-gon inscribed in a circle of radius R. This algebraic number s(n), n>=1, has the degree gamma(n) := A055035(n), and the row length of this table is gamma(n) + 1.

s(n) has been given in the power basis of the relevant algebraic number field in A228783 for even n (bisected into n == 0 (mod 4) and n == 2 (mod 4)), and in A228785 for odd n.

For the computation of the minimal polynomials ps(n,x), using the coefficients of s(n) in the relevant number field, and the conjugates of the corresponding algebraic numbers rho (giving the length ratios (smallest diagonal)/side in the relevant regular polygons see a comment on A228781. Note that the product of the gamma(n) linear factors (x - conjugates) has to be computed modulo the minimal polynomial of the relevant rho(k) = 2*cos(Pi/k) (called C(k,x=rho(k)) in A187360).

Thanks go to Seppo Mustonen who asked a question about the square of the sum of all length in the regular n-gon which led to this computation of s(n) and its minimal polynomial.

It would be interesting to find out which length ratios in the regular n-gon give the other positive zeros of the minimal polynomial ps(n,x). See some examples below.

LINKS

Table of n, a(n) for n=1..111.

FORMULA

a(n,m) = [x^m](minimal polynomial ps(n,x) of 2*sin(Pi/n) over the rationals), n>=1, m=0, ..., gamma(n), with gamma(n) = A055035(n).

EXAMPLE

The table a(n,m) starts:

n\m   0  1    2 3   4 5     6 7   8 9   10 12  13 14  15 16 17

1:    0  1

2:   -2  1

3:   -3  0    1

4:   -2  0    1

5:    5  0   -5 0   1

6:   -1  1

7:   -7  0   14 0  -7 0     1

8:    2  0   -4 0   1

9:   -3  0    9 0  -6 0     1

10:  -1  1    1

11: -11  0   55 0 -77 0    44 0 -11 0    1

12:   1  0   -4 0   1

13:  13  0  -91 0 182 0  -156 0  65 0  -13 0   1

14:   1 -2   -1 1

15:   1  0   -8 0  14 0    -7 0   1

16:   2  0  -16 0  20 0    -8 0   1

17:  17  0 -204 0 714 0 -1122 0 935 0 -442  0 119  0 -17  0  1

18:   1 -3    0 1

...

n = 19: [-19, 0, 285, 0, -1254, 0, 2508, 0, -2717, 0, 1729, 0, -665, 0, 152, 0, -19, 0, 1],

n = 20:  [1, 0, -12, 0, 19, 0, -8, 0, 1]

n = 5:  ps(5,x) = 5 -5*x^2 +1*x^4, with the zeros s(5) = sqrt(3 - tau), sqrt(2 + tau) =  tau*s(5) and their negative  values, where tau =rho(5) is the golden section. tau*s(5) is the length ratio diagonal/radius in the pentagon.

n = 7: ps(7,x) = -7 + 14*x^2 -7*x^4 + 1*x^6, with the positive zeros s(7) (side/R) about 0.868, s(7)*rho(7) (smallest diagonal/R) about 1.564, and s(7)*(rho(7)^2-1) (longer diagonal/R) about 1.950 in the heptagon inscribed in a circle with radius R.

n = 8:  ps(8,x) = 2 -4*x^2 + x^4, with the positive zeros s(8) = sqrt(2-sqrt(2)) and rho(8) = sqrt(2+sqrt(2)) (smallest diagonal/side).

n = 10: ps(10,x) =  -1 + x + x^2 with the positive zero s(10) = tau - 1 (the negative solution is -tau).

CROSSREFS

Cf. A187360, A055035, A228781, A228783, A228785.

Sequence in context: A071431 A277322 A182740 * A140699 A140256 A126206

Adjacent sequences:  A228783 A228784 A228785 * A228787 A228788 A228789

KEYWORD

sign,tabf

AUTHOR

Wolfdieter Lang, Oct 07 2013

STATUS

approved

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)