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A230079
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Table a(n,m) of coefficients of inverses of rho(A230078(n)), n>=2, with rho(k):= 2*cos(Pi/k), in the power basis of Q(rho(A230078(n))).
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3
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1, -1, 1, 2, 1, -1, -3, 0, 1, 3, 3, -4, -1, 1, 0, 4, 0, -1, -3, 6, 4, -5, -1, 1, 4, 4, -1, -1, -4, 10, 10, -15, -6, 7, 1, -1, 5, 10, -20, -15, 21, 7, -8, -1, 1, 0, 12, 0, -19, 0, 8, 0, -1, -8, -8, 6, 6, -1, -1, 6, 15, -35, -35, 56, 28, -36, -9, 10, 1, -1
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OFFSET
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2,4
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COMMENTS
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The length of row n is delta(A230078(n)), n>=2, with delta(k) = A055034(k).
The power base of the algebraic number field Q(rho(k)), with rho(k):= 2*cos(Pi/k), k >= 2, is <1, rho(k), rho(k)^2, ..., rho(k)^(delta(k)-1)>. Q(rho(k))-integers have integer coefficients in this basis. A230078(n), n >= 2, gives precisely the numbers k for which the inverse 1/rho(k) is a Q(rho(k))-integer. The present table a(n,m) lists these integer coefficients for 1/rho(A230078(n)), n >= 2, m = 0, 1, ..., delta(A230078(n))-1. delta(k) is the degree of the minimal polynomial C(k, x) of rho(k) (see A187360).
In general, 1/rho(k) = -(sum(c(k, m+1)*rho(k)^m, m=0..delta(k)-1))/c(k, 0), k >= 2, with the coefficients c(k ,m) of the minimal polynomial C(k, x) given in A187360(k, m). c(k ,0) = C(k, x=0) is +1 or -1 if and only if k is from {A230078(n), n>=2}, leading to a Q(rho(k))-integer.
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LINKS
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FORMULA
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a(n,m) = -c(b(n), m+1)/c(b(n), 0), with b(n) := A230078(n), for n>=2 and m= 0, 1, ... , delta(b(n)) -1, with delta(k) = A055034(k), and c(k, m) = A187360(k, m) (see a comment above on the minimal C polynomials).
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EXAMPLE
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The table a(n,m) begins, with b(n):=A230078(n):
n, b(n)\m 0 1 2 3 4 5 6 7 8 9 10 ...
2, 3: 1
3, 5: -1 1
4, 7: 2 1 -1
5, 9: -3 0 1
6, 11: 3 3 -4 -1 1
7, 12: 0 4 0 -1
8, 13: -3 6 4 -5 -1 1
9, 15: 4 4 -1 -1
10, 17: -4 10 10 -15 -6 7 1 -1
11, 19: 5 10 -20 -15 21 7 -8 -1 1
12, 20: 0 12 0 -19 0 8 0 -1
13, 21: -8 -8 6 6 -1 -1
14, 23: 6 15 -35 -35 56 28 -36 -9 10 1 -1
15, 24: 0 16 0 -20 0 8 0 -1
...
n=2: C(3, x) = x - 1, delta(3) =1, 1/rho(3) = 1, a rational integer.
n=3: C(5, x) =x^2 - x -1, delta(5) = 2, a(3,0) = - c(5, 1)/c(5, 0) = -(-1)/(-1) = -1, a(3,1) = - c(5, 2)/c(5, 0) = -1/(-1) = +1.
n =3: rho(5) = tau := (1 + sqrt(5))/2 (golden section); 1/rho(5) = -1*1 + 1*rho(5).
n= 4: rho(7) = 2*cos(Pi/7), (approximately 1.801937736); 1/rho(7) = 2*1 + 1*rho(7) - 1*rho(7)^2, (approximately 0.5549581320).
n=10: rho(17) = 2*cos(Pi/17), (approximately 1.965946199); 1/rho(17) = -4*1 + 10*rho(17) + 10*rho(17)^2 - 15*rho(17)^3 - 6*rho(17)^4 + 7*rho(17)^5 + 1*rho(17)^6 -1*rho(17)^7, (approximately 0.5086609190).
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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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