This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A228786 #27 Jun 01 2024 09:58:05
%S A228786 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,
%T A228786 -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,
%U A228786 -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
%N A228786 Table of coefficients of the minimal polynomials of 2*sin(Pi/n), n >= 1.
%C A228786 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.
%C A228786 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.
%C A228786 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).
%C A228786 Thanks go to _Seppo Mustonen_, who asked a question about the square of the sum of all lengths in the regular n-gon, which led to this computation of s(n) and its minimal polynomial.
%C A228786 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.
%C A228786 The zeros of the row polynomials ps(n,x) are 2*cos(2*Pi*k/c(2*n))) for gcd(k, c(2*n)) = 1, where c(n) = A178182(n), and k from {0, ..., floor(c(2*n)/2)}, for n >= 1. The number of these solutions is gamma(n) = A055035(n). See the formula section. This results from the zeros of the minimal polynomials of sin(2*Pi/n), with coefficients given in A181872/A181873. - _Wolfdieter Lang_, Oct 30 2019
%F A228786 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).
%F A228786 ps(n,x) = Product_{k=0..floor(c(2*n)/n) and gcd(k, c(2*n)) = 1} (x - 2*cos(2*Pi*k/c(2*n)), with c(2*n) = A178182(2*n), for n >= 1. There are gamma(n) = A055035(n) zeros. - _Wolfdieter Lang_, Oct 30 2019
%e A228786 The table a(n, m) starts:
%e A228786 n\m 0 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17
%e A228786 1: 0 1
%e A228786 2: -2 1
%e A228786 3: -3 0 1
%e A228786 4: -2 0 1
%e A228786 5: 5 0 -5 0 1
%e A228786 6: -1 1
%e A228786 7: -7 0 14 0 -7 0 1
%e A228786 8: 2 0 -4 0 1
%e A228786 9: -3 0 9 0 -6 0 1
%e A228786 10: -1 1 1
%e A228786 11: -11 0 55 0 -77 0 44 0 -11 0 1
%e A228786 12: 1 0 -4 0 1
%e A228786 13: 13 0 -91 0 182 0 -156 0 65 0 -13 0 1
%e A228786 14: 1 -2 -1 1
%e A228786 15: 1 0 -8 0 14 0 -7 0 1
%e A228786 16: 2 0 -16 0 20 0 -8 0 1
%e A228786 17: 17 0 -204 0 714 0 -1122 0 935 0 -442 0 119 0 -17 0 1
%e A228786 18: 1 -3 0 1
%e A228786 ...
%e A228786 n = 19: [-19, 0, 285, 0, -1254, 0, 2508, 0, -2717, 0, 1729, 0, -665, 0, 152, 0, -19, 0, 1],
%e A228786 n = 20: [1, 0, -12, 0, 19, 0, -8, 0, 1]
%e A228786 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.
%e A228786 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.
%e A228786 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).
%e A228786 n = 10: ps(10,x) = -1 + x + x^2 with the positive zero s(10) = tau - 1 (the negative solution is -tau).
%Y A228786 Cf. A187360, A055035, A228781, A228783, A228785.
%K A228786 sign,tabf
%O A228786 1,3
%A A228786 _Wolfdieter Lang_, Oct 07 2013