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A232630
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Coefficient table for the minimal polynomials of 2*sin(4*Pi/n). Rising powers of x.
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1
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0, 1, 0, 1, -3, 0, 1, 0, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 14, 0, -7, 0, 1, -2, 1, -3, 0, 9, 0, -6, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, -3, 0, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, -7, 0, 14, 0, -7, 0, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, -2, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1
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OFFSET
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1,5
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COMMENTS
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The length of row n is A232626(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 2, 7, 5, 11, 3, 13, 7, 9, 3, 17, 7, 19, 5,...
In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2.
For the power basis representation of 2*sin(4*Pi/n) in the algebraic number field Q(rho(q(2,n))), with q(2,n)) = A232625(n) and rho(m) := 2*cos(Pi/m), see A232629. Call the row polynomials of A232629 PB2(n,x) (power basis polynomial for the case k=2 in 2*sin(2*Pi*k/n)).
The minimal polynomial of 2*sin(4*Pi/n), call it MP2(n, x), is obtained from the conjugates rho(q(2,n),j), j= 1, ... , delta(q(2,n)) = A232626(n), which are the zeros of C(q(2,n), x), the minimal polynomial of rho(q(2,n)) = rho(q(2,n),1) (for C see A187360). MP2(n, x) = product(x - PB2(n, rho(q(2,n),j)), j=1..A232626(n)) (mod C(q(2,n), rho(q(2,n)))).
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LINKS
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FORMULA
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a(n,m) = [x^m] MP2(n, x), n>=1, m = 0, 1, ..., A232626(n), with the minimal polynomials of 2*sin(4*Pi/n), computed like explained above in a comment.
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EXAMPLE
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The table a(n,m) begins:
--------------------------------------------------------------------------------------
n\m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
1: 0 1
2: 0 1
3: -3 0 1
4: 0 1
5: 5 0 -5 0 1
6: -3 0 1
7: -7 0 14 0 -7 0 1
8: -2 1
9: -3 0 9 0 -6 0 1
10: 5 0 -5 0 1
11: -11 0 55 0 -77 0 44 0 -11 0 1
12: -3 0 1
13: 13 0 -91 0 182 0 -156 0 65 0 -13 0 1
14: -7 0 14 0 -7 0 1
15: 1 0 -8 0 14 0 -7 0 1
16: -2 0 1
17: 17 0 -204 0 714 0 -1122 0 935 0 -442 0 119 0 -17 0 1
18: -3 0 9 0 -6 0 1
19: -19 0 285 0 -1254 0 2508 0 -2717 0 1729 0 -665 0 152 0 -19 0 1
20: 5 0 -5 0 1
...
n=1: 2*sin(4*Pi/1) = 0 is rational, therefore MP2(1, x) = x, with coefficients 0, 1, and degree A232626(1) = 1. PB2(1, rho(1,1)) = PB2(1, rho(1)) = 0.
n=3: A232626(2) = 2. PB2(2, x) = -x, C(6, x) = x^2 - 3, with zeros rho(6) and R(5, rho(6)) (for R see A127672), hence rho(6,1) = rho(6) and rho(6,2) = R(5, rho(6))= 5*rho(6) - 5*rho(6)^3 + 1*rho(6)^5, MP2(3, x) = (x - (-rho(6)))*(x - (- R(5, rho(6))) reduced with rho(6)^2 = 3 leading to MP2(3, x) = -3 + x^2, yielding row n=3: -3 0 1.
n=8: this row -2, 1 coincides with row n=4 of A231188.
n=17: coincides with WolframAlpha's MinimalPolynomial[2*sin(4*Pi/17),x] = 17-204 x^2+714 x^4-1122 x^6+935 x^8-442 x^10+119 x^12-17 x^14+x^16.
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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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