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A232631
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Coefficient table for minimal polynomials of s(2*l)^2 = (2*sin(Pi/(2*l)))^2.
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4
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-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, 1, 1, -12, 19, -8, 1, -1, 15, -35, 28, -9, 1, 1, -16, 20, -8, 1, 1, -21, 70, -84, 45, -11, 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
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OFFSET
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1,1
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COMMENTS
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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, ...
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)).
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.
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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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The table a(l,m) begins (n = 2*l):
n, l\m 0 1 2 3 4 5 6 ...
2 , 1: -4 1
4, 2: -2 1
6, 3: -1 1
8, 4: 2 -4 1
10, 5: 1 -3 1
12, 6: 1 -4 1
14, 7: -1 6 -5 1
16, 8: 2 -16 20 -8 1
18, 9: -1 9 -6 1
20, 10: 1 -12 19 -8 1
22, 11: -1 15 -35 28 -9 1
24, 12: 1 -16 20 -8 1
26, 13: 1 -21 70 -84 45 -11 1
28, 14: 1 -24 86 -104 53 -12 1
30, 15: 1 -24 26 -9 1
...
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.
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.
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MATHEMATICA
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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
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CROSSREFS
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KEYWORD
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sign,tabf,easy
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AUTHOR
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STATUS
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approved
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