cp's OEIS Frontend

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.

%I A334429 #27 Sep 12 2023 08:28:48
%S A334429 -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,
%T A334429 15,-35,28,-9,1,1,-4,1,1,-21,70,-84,45,-11,1,-7,14,-7,1,1,-24,26,-9,1,
%U A334429 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
%N 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.
%C A334429 The length of row n is A023022(n) + 1, with A023022(1) = 1.
%C A334429 For the minimal polynomials of 2*sin(Pi/n) see A232633 (n >= 1), A232632 (even n) and A232631 (odd n).
%C A334429 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).
%C A334429 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.
%C A334429 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.
%C A334429 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.
%C A334429 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.
%C A334429 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_.
%H A334429 Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(Pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018v2 [math.GR], 2012.
%H A334429 Jesús Salas and Alan D. Sokal, <a href="https://arxiv.org/abs/cond-mat/0004330">Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models I. General Theory and Square-Lattice Chromatic Polynomial</a>, 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.
%F A334429 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.
%e A334429 The irregular triangle T(n, k) begins:
%e A334429 n\k   0    1    2    3     4     5     6    7    8  9 ...
%e A334429 1:   -4    1
%e A334429 2:    0    1
%e A334429 3:   -1    1
%e A334429 4;   -2    1
%e A334429 5:    1   -3    1
%e A334429 6:   -3    1
%e A334429 7:   -1    6   -5    1
%e A334429 8:    2   -4    1
%e A334429 9:   -1    9   -6    1
%e A334429 10:   5   -5    1
%e A334429 11:  -1   15  -35   28    -9     1
%e A334429 12:   1   -4    1
%e A334429 13:   1  -21   70  -84    45   -11     1
%e A334429 14:  -7   14   -7    1
%e A334429 15:   1  -24   26   -9     1
%e A334429 16:   2  -16   20   -8     1
%e A334429 17:   1  -36  210 -462   495  -286    91  -15    1
%e A334429 18:  -3    9   -6    1
%e A334429 19:  -1   45 -330  924 -1287  1001  -455  120  -17  1
%e A334429 20:   1  -12   19   -8
%e A334429 ...
%Y A334429 Cf. A023022, A187360, A232631, A232632, A232633, A334431 (even n), A334432 (odd n).
%K A334429 sign,tabf,easy
%O A334429 1,1
%A A334429 _Wolfdieter Lang_, Jun 15 2020