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%I A232631 #13 Aug 10 2025 10:33:26
%S A232631 -4,1,-2,1,-1,1,2,-4,1,1,-3,1,1,-4,1,-1,6,-5,1,2,-16,20,-8,1,-1,9,-6,
%T A232631 1,1,-12,19,-8,1,-1,15,-35,28,-9,1,1,-16,20,-8,1,1,-21,70,-84,45,-11,
%U A232631 1,1,-24,86,-104,53,-12,1,1,-24,26,-9,1,2,-64,336,-672,660,-352,104,-16,1,1,-36,210,-462,495,-286,91,-15,1
%N A232631 Coefficient table for minimal polynomials of s(2*l)^2 = (2*sin(Pi/(2*l)))^2.
%C A232631 The length of row l of this table is delta(l) + 1 = A055034(l) + 1, l >= 1, that is: 2, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 5, ...
%C A232631 s(n):= 2*sin(Pi/n) is the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some length units). In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), the length ratio (smallest diagonal)/s(n) in the regular n-gon (n>=2). 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 s(2*l) is an integer in the algebraic number field Q(rho(l)).
%C A232631 Its (monic) minimal polynomial is obtained from the conjugates of rho(l), called rho(l;j), j = 1, 2, ..., delta(l), which are the zeros of the minimal polynomial of rho(l) = rho(l;1) of degree delta(l) = A055034(l), called C(l, x) in A187360. These conjugates are therefore rho(l;j) = 2*cos(Pi*rpnodd(l,j)/l) where rpnodd(l,j) is the j-th entry of the list rpnodd(l) of the odd numbers < l which are relatively prime to l (for example, rpnodd(9) = [1,5,7], and rpnodd(9,2) = 5). From this the conjugates of s(2*l)^2 become 2 - rho(l;j), and the minimal polynomial of s(2*l)^2 is MPs2(l, x) = product( x - (2- rho(l;j)), j=1..delta(l)) for l >=1. Because the zeros of C(l, x) are integers in the algebraic number field Q(rho(l)) written in its power basis (see table 4 of the link under A187360 to the Q(2 cos(Pi/n)) paper) one finds, after expansion and reducing powers of rho(l) modulo C(l, rho(l)), directly the integer coefficients appropriate for this (monic) minimal polynomial. Only the equation C(l, rho(l)) = 0 is needed, not the trigonometric version of rho(l) and its powers.
%C A232631 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.
%F A232631 a(l,m) = [x^m] MPs2(l, x), l >= 1, m = 0, 1, ...., delta(l), with the minimal polynomial MPs2(l, x) of (2*sin(Pi/(2*l)))^2, given above in a comment. The degree delta(l) = A055034(l).
%e A232631 The table a(l,m) begins (n = 2*l):
%e A232631 n, l\m 0 1 2 3 4 5 6 ...
%e A232631 2, 1: -4 1
%e A232631 4, 2: -2 1
%e A232631 6, 3: -1 1
%e A232631 8, 4: 2 -4 1
%e A232631 10, 5: 1 -3 1
%e A232631 12, 6: 1 -4 1
%e A232631 14, 7: -1 6 -5 1
%e A232631 16, 8: 2 -16 20 -8 1
%e A232631 18, 9: -1 9 -6 1
%e A232631 20, 10: 1 -12 19 -8 1
%e A232631 22, 11: -1 15 -35 28 -9 1
%e A232631 24, 12: 1 -16 20 -8 1
%e A232631 26, 13: 1 -21 70 -84 45 -11 1
%e A232631 28, 14: 1 -24 86 -104 53 -12 1
%e A232631 30, 15: 1 -24 26 -9 1
%e A232631 ...
%e A232631 The minimal polynomial of s(10)^2 = (2*sin(Pi/10))^2 = 2 - rho(5) is MPs2(5, x) = product(x - (2- rho(5;j)), j=1..2) = (x - (2 - phi))*(x - (2 - (1-phi))) with rho(5) = phi the golden section satisfying C(5, phi) = phi^2 - phi -1 = 0, hence MPs2(5, x) = 2 + phi - phi^2 - 3*x + x^2 = 1 - 3*x + x^2.
%e A232631 The row n=26 checks with WolframAlpha's MinimalPolynomial[(2*sin(Pi/26))^2 ,x] = 1-21 x+70 x^2-84 x^3+45 x^4-11 x^5+x^6.
%t A232631 Flatten[ CoefficientList[ Table[ MinimalPolynomial[ (2*Sin[Pi/(2*l)])^2, x], {l, 1, 17}], x]] (* adapted from _Jean-François Alcover_, A187360 *) (* _Wolfdieter Lang_, Dec 23 2013 *)
%Y A232631 Cf. A232632 (odd n), A232633 (all n), A055034 (degree).
%K A232631 sign,tabf,easy
%O A232631 1,1
%A A232631 _Wolfdieter Lang_, Dec 18 2013