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A232624 Coefficient table for the minimal polynomials of 2*cos(2*Pi/n). 5
-2, 1, 2, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -2, 1, 1, -2, 0, 1, 1, -3, 0, 1, -1, -1, 1, 1, 3, -3, -4, 1, 1, -3, 0, 1, -1, 3, 6, -4, -5, 1, 1, 1, -2, -1, 1, 1, 4, -4, -1, 1, 2, 0, -4, 0, 1, 1, -4, -10, 10, 15, -6, -7, 1, 1, -1, -3, 0, 1, 1, 5, -10, -20, 15, 21, -7, -8, 1, 1, 5, 0, -5, 0, 1, 1, -8, 8, 6, -6, -1, 1, -1, 3, 3, -4, -1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The length of row n is deg(n) + 1, n >= 1, with the degree deg(1) = deg(2) = 1, and deg(n) = phi(n)/2 = A023022(n) for n >= 3. That is: 2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4, 10, 5, ...

2*cos(2*Pi/n) = R(2, rho(n)) = -2 + rho(n)^2, with rho(n) = 2*cos(Pi/n) and the monic Chebyshev T-polynomials R(n, x), n>=1, with coefficient table A127672. For even n 2*cos(2*Pi/n) becomes rho(n/2). Therefore, 2*cos(2*Pi/n) is an integer in the algebraic number field Q(rho(n/2)) or Q(rho(n)) if n is even or odd, respectively. The degree deg(n) of the minimal polynomials, call them MPR2(n, x), is delta(n/2) or delta(n) for even or odd n, respectively, with delta(n) = A055034(n). This becomes deg(n) as given above.

These minimal polynomials are C(n/2, x) if n is even, with C(k, x) the minimal polynomials of rho(k) given in A187360.

For odd n the known zeros of C(n, x) are rho(n) and its conjugates, call them rho(n;j), j=1, 2, ..., delta(n), with rho(n;1) = rho(n). These conjugates can be written in the power basis of Q(rho(2*l+1)), l >= 1. See the link to the Q(2cos(Pi/n)) paper in A187360, and there Table 4. Then the (monic) minimal polynomial MPR2(2*l+1, x) = Product_{j=1..delta(2*l+1)} (x - (-2 + rho(2*l+1;j)^2)), l >= 0. After expansion all powers of rho(2*l+1) not smaller than delta(2*l+1) are reduced with the help of C(2*l+1,rho(2*l+1)) = 0, leading automatically to integer coefficients (without using the trigonometric version of rho(2*l+1)).

Compare the present minimal polynomials with the (non-monic) minimal polynomials of cos(2*Pi/n) given in an Artur Jasinski comment from Oct 28 2008 on A023022.

The present monic integer minimal polynomials of 2*cos(2*Pi/n), called MPR2(n, x), are related to the non-monic integer minimal polynomials of 2*cos(2*Pi/n) of A181877, called there psi(n, x) by MPR2(n, x) = psi(n, x/2). See Table 5 of the Wolfdieter Lang link given there. - Wolfdieter Lang, Nov 29 2013

The present minimal polynomials MPR2(n, x) are C(n/2, x) if n is even (see above) and (-1)^degC(n)*C(n, -x) if n is odd, with the C polynomials from A187360 of degree degC(n) = A055034(n). Note that degC(2*k+1) = deg(2*k+1) = A023022(2*k+1), k >= 0. - Wolfdieter Lang, Apr 12 2018

Let {U(n, x)} be defined as: U(0, x) = 0, U(1, x) = 1, U(n, x) = x*U(n-1, x) - U(n-2, x) for n >= 2, then U(n, x) = Product_{k|2n, k>=3} MPR2(k, x) for n > 0, because U(n, x) = Product_{m=1..n-1} (x - 2*cos(Pi*m/n)) for n > 0. - Jianing Song, Jul 08 2019

Conjecture: For odd n > 1, the term of the highest degree of (MPR2(2n, x) - MPR2(n, x))/2 is (-1)^omega(n) * x^(phi(n)/2-n/rad(n)) = A076479(n) * x^(A023022(n)-A003557(n)). For example, for n = 15, (MPR2(30, x) - MPR2(15, x))/2 = x^3 - 4x; for n = 105, (MPR2(210, x) - MPR2(105, x))/2 = -x^23 + ...; for n = 225, (MPR2(450, x) - MPR2(225, x))/2 = x^45 + ... If this is true, then for odd n > 1, a(n,A023022(n)-k) = a(2n,A023022(n)-k) = 0 for k = 1, 3, ..., A003557(n)-2; a(n,A023022(n)-A003557(n)) = -A076479(n) and a(2n,A023022(n)-A003557(n)) = A076479(n). - Jianing Song, Jul 11 2019

LINKS

Table of n, a(n) for n=1..98.

FORMULA

a(n,m) = [x^m] MPR2(n, x), n >= 1, m=0, 1, ..., deg(n), with MPR2(n, x) the (monic) minimal polynomials of 2*cos(2*Pi/n), explained in a comment above. The degree is deg(1) = deg(2) = 1, deg(n) = phi(n)/2 = A023022(n), n >= 3 (phi is the Euler totient function A000010).

From Jianing Song, Jul 09 2019: (Start)

MPR2(n, x) = Product_{0<=m<=n/2, gcd(m, n)=1} (x - 2*cos(2Pi*m/n)).

If 4 divides n, then MPR2(n, x) = Product_{k|(n/2)} U((n/2)/k, x)^mu(k), where U(n, x) is the polynomial defined in comment and mu = A008683. For odd n, MPR2(n, x)*MPR2(2n, x) = Product_{k|n} U(n/k, x)^mu(k).

If 4 divides n and n > 4, then a(n,2k+1) = 0, that is, MPR2(n, x) contains even powers of x only.

For odd n > 1, a(2n,k) = a(n,k)*(-1)^(A023022(n)-k). (End)

EXAMPLE

The table a(n,m) begins:

n\m   0   1    2    3    4    5   6   7   8   9 ...

1:   -2   1

2:    2   1

3:    1   1

4:    0   1

5:   -1   1    1

6:   -1   1

7:   -1  -2    1    1

8:   -2   0    1

9:    1  -3    0    1

10:  -1  -1    1

11:   1   3   -3   -4    1    1

12:  -3   0    1

13:  -1   3    6   -4   -5    1   1

14:   1  -2   -1    1

15:   1   4   -4   -1    1

16:   2   0   -4    0    1

17:   1  -4  -10   10   15   -6  -7   1   1

18:  -1  -3    0    1

19:   1   5  -10  -20   15   21  -7  -8   1   1

20:   5   0   -5    0    1

...

MPR2(14, x) = C(7, x) = 1  - 2*x  - x^2  + x^3.

MPR2(7, x) = (x - (-2 + z^2))*(x - (-2 + (-1 - z + z^2)^2))*

  (x - (-2 + (2 - z^2)^2)), with z = rho(7). Expanded and reduced with C(7, z) = 0 this becomes finally MPR2(7, x) = -1 - 2*x + x^2 + x^3.

MPR2(7, x) = -C(7, -x). - Wolfdieter Lang, Apr 12 2018

MATHEMATICA

ro[n_] := (MPR2 = CoefficientList[p = MinimalPolynomial[2*Cos[2*(Pi/n)], x], x]; MPR2); Flatten[Table[ro[n], {n, 1, 30}]] (* Jianing Song, Jul 09 2019 *)

CROSSREFS

Cf. A023022 (degree), A055034, A187360 (C polynomials).

Cf. A181877, A181875/A181876. - Wolfdieter Lang, Nov 29 2013

Sequence in context: A161780 A136571 A178562 * A287368 A108149 A128583

Adjacent sequences:  A232621 A232622 A232623 * A232625 A232626 A232627

KEYWORD

sign,tabf,easy

AUTHOR

Wolfdieter Lang, Nov 28 2013

STATUS

approved

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Last modified July 11 15:58 EDT 2020. Contains 335626 sequences. (Running on oeis4.)