OFFSET
1,3
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(x - 2*(1 + cos(Pi*(2*j+1)/p)), j=0..(p-3)/2), 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.
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.
FORMULA
EXAMPLE
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(x - 2*(1 + cos(Pi*(2*j+1)/7)), j=0..2) = (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(x - 2*(1 - cos(Pi*(2*j+1)/7)), j=0..2) = (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)).
MATHEMATICA
Flatten[ CoefficientList[ Table[ MinimalPolynomial[ (2*Sin[Pi/n])^2, x], {n, 1, 20}], x]] (adapted from Jean-François Alcover, A187360) - Wolfdieter Lang, Dec 24 2013
CROSSREFS
KEYWORD
sign,tabf,easy,changed
AUTHOR
Wolfdieter Lang, Dec 19 2013
STATUS
approved