A334429 Irregular triangle read by rows: T(n, k) gives the coefficients of x^k of the minimal polynomials of the algebraic number over the rationals rho(n)^2, with rho(n) = 2*cos(Pi/n), for n >= 1.

-4, 1, 0, 1, -1, 1, -2, 1, 1, -3, 1, -3, 1, -1, 6, -5, 1, 2, -4, 1, -1, 9, -6, 1, 5, -5, 1, -1, 15, -35, 28, -9, 1, 1, -4, 1, 1, -21, 70, -84, 45, -11, 1, -7, 14, -7, 1, 1, -24, 26, -9, 1, 2, -16, 20, -8, 1, 1, -36, 210, -462, 495, -286, 91, -15, 1, -3, 9, -6, 1, -1, 45, -330, 924, -1287, 1001, -455, 120, -17, 1, 1, -12, 19, -8, 1

Offset: 1

Author

Wolfdieter Lang, Jun 15 2020

Keywords

• sign
• tabf
• easy

Comments

The length of row n is A023022(n) + 1, with A023022(1) = 1.

For the minimal polynomials of 2*sin(Pi/n) see A232633 (n >= 1), A232632 (even n) and A232631 (odd n).

The degree of the algebraic number over the rationals rho(n) = 2*cos(Pi/n) is delta(n) = A055034(n). The degree of rho(n)^2, for n = 2*m, is delta(m), for m >= 1. This is due to the trigonometric identity (half-angle formula) rho(2*m)^2 = 2 + rho(m). For m >= 0 rho(2*m+1)^2 has degree delta(2*l+1).

For the field extension Q(rho(n)) see the W. Lang link where the minimal polynomial of rho(n), named C(n, x), is shown in Table 2. See also A187360.

In both cases the conjugates (over Q) of rho(n), that is the roots of the minimal polynomial C(n, x) enter. This is the set with elements 2*cos(Pi*(2*m+1)/n) = R(2*m+1, rho(n)), for m from {0..floor((n-1)/2)} with gcd(2*m + 1, n) = 1. The polynomial R(n, x) = 2*T(n, x/2) is a monic version of the Chebyshev T polynomials; see A127672 for its coefficients. This list of numbers 2*m+1 is named rpnodd(n) (e.g., n = 12, rpnodd(n) = [1, 5, 7, 11]). #rpnodd(n) = delta(n). The conjugates of rho(n) are then rho(n; j) = 2*cos(Pi*rpnodd(n)_j/n), for j = 1, 2, ..., delta(n), and rho(n; 1) = rho(n), for n >= 2. Because rpnodd(1) is the empty set, a separate case is needed, namely rho(1; 1) = -2.

The minimal polynomials for rho(2*m)^2 are then MPc2(m, x) = Product_{j=1..delta(m)} (x - (2 + rho(m; j)) = Product_{j=1..delta(m)} (x - (2 + R(rpnodd(m)_j, rho(m)))) for m >= 2, and MPc2(1, x) = x. But because C(m, rho(m)) = 0, this has to be evaluated modulo this minimal polynomial of rho(m), that is all powers rho(m)^k with k >= delta(m) are replaced, leaving elements of Q(rho(m)) written in its power basis. Note that the trigonometric form of rho(m) is not used in this computation.

In the odd n case one uses for the conjugates of rho(2*m+1)^2 the formula R(2*m+1, x)^2 = R(2*(2*m+1), x) + 2, obtained from the product formula for R(n, x)*R(k, x) = R(n+m, x) + 2. Then for the reduced 2*m+1 values defined above R(2*(2*m+1), x) + 2 can be replaced by -R(rpnodd(2*m+1)j, x) + 2, for j = 1, ..., delta(2*m+1). Thus MPc2(2*m+1, x) = Product{j=1..delta(2*m+1)} (x - (2 - R(rpnodd(2*m+1)_j, x)), for m >= 1. But for m = 0 (n = 1) the degree of rho(1)^2 = (-2)^2 is 1, hence MPc2(1, x) = x - 4.

These polynomials appear, e.g., in the Salas and Sokal paper, see Table 1, p. 64, or p. 620, for n = 2..16, where rho(n)^2 are called Beraha numbers B_n. I was informed about this paper by Gary W. Adamson.

Examples

			The irregular triangle T(n, k) begins:
n\k   0    1    2    3     4     5     6    7    8  9 ...
1:   -4    1
2:    0    1
3:   -1    1
4;   -2    1
5:    1   -3    1
6:   -3    1
7:   -1    6   -5    1
8:    2   -4    1
9:   -1    9   -6    1
10:   5   -5    1
11:  -1   15  -35   28    -9     1
12:   1   -4    1
13:   1  -21   70  -84    45   -11     1
14:  -7   14   -7    1
15:   1  -24   26   -9     1
16:   2  -16   20   -8     1
17:   1  -36  210 -462   495  -286    91  -15    1
18:  -3    9   -6    1
19:  -1   45 -330  924 -1287  1001  -455  120  -17  1
20:   1  -12   19   -8
...
		

Links

• Wolfdieter Lang, The field Q(2cos(Pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018v2 [math.GR], 2012.
• Jesús Salas and Alan D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial, arXiv:cond-mat/0004330v2 [cond-mat.stat-mech], 2001, and J. Stat. Phys. 104, Nos. 3/4, (2001) 609-699, Table 1 on p. 620.

Crossrefs

Cf. A023022, A187360, A232631, A232632, A232633, A334431 (even n), A334432 (odd n).

Formula

T(n, k) = [x^k] MPc2(n, x), for n >= 1, and k = 0, 1, 2, ..., A023022(n), with A023022(1) = 1. For the Mpc2(n, x) formulas for even and odd n see the comments above.