A232633 Coefficient table for minimal polynomials of s(n)^2 = (2*sin(Pi/n))^2.

0, 1, -4, 1, -3, 1, -2, 1, 5, -5, 1, -1, 1, -7, 14, -7, 1, 2, -4, 1, -3, 9, -6, 1, 1, -3, 1, -11, 55, -77, 44, -11, 1, 1, -4, 1, 13, -91, 182, -156, 65, -13, 1, -1, 6, -5, 1, 1, -8, 14, -7, 1, 2, -16, 20, -8, 1, 17, -204, 714, -1122, 935, -442, 119, -17, 1, -1, 9, -6, 1, -19, 285, -1254, 2508, -2717, 1729, -665, 152, -19, 1

Offset: 1

Author

Wolfdieter Lang, Dec 19 2013

Keywords

• sign
• tabf
• easy

Comments

The length of row n of this table is 1 + A023022(n), n >= 0, that is 2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4,...

s(n):= 2*sin(Pi/n) is for n >= 2 the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some units). s(1) = 0. In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), for n>=2 this is the length ratio (smallest diagonal)/s(n) in the regular n-gon. If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) +2). Therefore, if n is even s(n)^2 is an integer in the algebraic number field Q(rho(n/2)), and if n is odd then it is an integer in Q(rho(n)). The coefficient tables for the minimal polynomials of s(n)^2, called MPs2(n, x), for even and odd n have been given in A232631 and A232632, respectively. See these entries for details, and the link to the Q(2 cos(pi/n)) paper, Table 4, in A187360 for the power basis representation of the zeros of the minimal polynomial C(n, x) of rho(n).

The degree deg(n) of MPs2(n, x) is therefore delta(n/2) or delta(n) for n even or odd, respectively, where delta(n) = A055034(n). This means that deg(1) = deg(2) =1 and deg(n) = phi(n)/2 = A023022(n), n >= 3. deg(n) = A023022(n).

Especially MPs2(p, x) = Product_{j=0..(p-3)/2} (x - 2*(1 + cos(Pi*(2*j+1)/p))), for p an odd prime (A065091).

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called ``Polynomials associated with squared diagonals of regular polygons'', Nov 16 2013.

Examples

			The table a(n,m) begins:
  n/m    0     1      2      3       4     5     6    7    8   9 ...
  1:     0     1
  2:    -4     1
  3:    -3     1
  4:    -2     1
  5:     5    -5      1
  6:    -1     1
  7:    -7    14     -7      1
  8:     2    -4      1
  9:    -3     9     -6      1
  10:    1    -3      1
  11:  -11    55    -77     44     -11     1
  12:    1    -4      1
  13:   13   -91    182   -156      65   -13     1
  14:   -1     6     -5      1
  15:    1    -8     14     -7       1
  16:    2   -16     20     -8       1
  17:   17  -204    714  -1122     935  -442   119  -17    1
  18:   -1     9     -6      1
  19:  -19   285  -1254   2508   -2717  1729  -665  152  -19   1
  20:    1   -12     19     -8       1
  ...
MPs2(7, x) = Product_{j=0..2} (x - 2*(1 + cos(Pi*(2*j+1)/7))) = (x - (2 + rho(7)))*(x - (2 + (-1 - rho(7) + rho(7)^2)))*(x - (2 + (2 - rho(7)^2))) = (-8+4*z-2*z^2-5*z^3+z^4+z^5) + (14-z+2*z^2+z^3-z^4)*x -7*x^2 +x^3, with z = rho(7), and this becomes  due to C(7, z) = z^3 - z^2 - 2*z + 1, finally MPs2(7, x) = -7 + 14*x - 7*x^2 + x^3.
MPs2(14, x) = Product_{j=0..2} (x - 2*(1 - cos(Pi*(2*j+1)/7))) = (x - (2 - rho(7)))*(x - (2 - (-1 - rho(7) + rho(7)^2)))*(x - (2 - (2 - rho(7)^2))) = -1 + 6*x - 5*x^2 + x^3 (using again C(7, z) = 0 with z = rho(7)).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..14000 (rows n = 1..300, flattened.)
• Johann Cigler and Hans-Christian Herbig, Factorization of spread polynomials, arXiv:2412.18958 [math.NT], 2024. See p. 6.

Crossrefs

Cf. A232631 (even n), A232632 (odd n), A023022 (degree), A187360.

Programs

Mathematica

Flatten[ CoefficientList[ Table[ MinimalPolynomial[ (2*Sin[Pi/n])^2, x], {n, 1, 19}], x]] (* adapted from Jean-François Alcover, A187360 *) (* Wolfdieter Lang, Dec 24 2013 *)

Formula

a(n,m) = [x^m] MPs2(n, x), n >= 0, m = 0, 1, ..., deg(n), with the minimal polynomial MPs2(n, x) of s(n)^2 = (2*sin(Pi/n))^2. The degree is deg(n) = A023022(n).

a(2*l,m) = A232631(l,m), l >= 1, a(2*l+1,m) = A232832(l,m), l >= 0.