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 A232633 #21 Aug 13 2025 23:14:04
%S A232633 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,
%T A232633 -11,55,-77,44,-11,1,1,-4,1,13,-91,182,-156,65,-13,1,-1,6,-5,1,1,-8,
%U A232633 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
%N A232633 Coefficient table for minimal polynomials of s(n)^2 = (2*sin(Pi/n))^2.
%C A232633 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,...
%C A232633 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).
%C A232633 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).
%C A232633 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).
%C A232633 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.
%H A232633 Michael De Vlieger, <a href="/A232633/b232633.txt">Table of n, a(n) for n = 1..14000</a> (rows n = 1..300, flattened.)
%H A232633 Johann Cigler and Hans-Christian Herbig, <a href="https://arxiv.org/abs/2412.18958">Factorization of spread polynomials</a>, arXiv:2412.18958 [math.NT], 2024. See p. 6.
%F A232633 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).
%F A232633 a(2*l,m) = A232631(l,m), l >= 1, a(2*l+1,m) = A232832(l,m), l >= 0.
%e A232633 The table a(n,m) begins:
%e A232633 n/m 0 1 2 3 4 5 6 7 8 9 ...
%e A232633 1: 0 1
%e A232633 2: -4 1
%e A232633 3: -3 1
%e A232633 4: -2 1
%e A232633 5: 5 -5 1
%e A232633 6: -1 1
%e A232633 7: -7 14 -7 1
%e A232633 8: 2 -4 1
%e A232633 9: -3 9 -6 1
%e A232633 10: 1 -3 1
%e A232633 11: -11 55 -77 44 -11 1
%e A232633 12: 1 -4 1
%e A232633 13: 13 -91 182 -156 65 -13 1
%e A232633 14: -1 6 -5 1
%e A232633 15: 1 -8 14 -7 1
%e A232633 16: 2 -16 20 -8 1
%e A232633 17: 17 -204 714 -1122 935 -442 119 -17 1
%e A232633 18: -1 9 -6 1
%e A232633 19: -19 285 -1254 2508 -2717 1729 -665 152 -19 1
%e A232633 20: 1 -12 19 -8 1
%e A232633 ...
%e A232633 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.
%e A232633 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)).
%t A232633 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 *)
%Y A232633 Cf. A232631 (even n), A232632 (odd n), A023022 (degree), A187360.
%K A232633 sign,tabf,easy
%O A232633 1,3
%A A232633 _Wolfdieter Lang_, Dec 19 2013