cp's OEIS Frontend

The length of row n of this table is 1 + A023022(n), n >= 0, that is 2, 2, 2, 2, 3, 2, 4, 3, 4, 3, 6, 3, 7, 4, 5, 5, 9, 4,...

s(n):= 2*sin(Pi/n) is for n >= 2 the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some units). s(1) = 0. In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), for n>=2 this is the length ratio (smallest diagonal)/s(n) in the regular n-gon. If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) +2). Therefore, if n is even s(n)^2 is an integer in the algebraic number field Q(rho(n/2)), and if n is odd then it is an integer in Q(rho(n)). The coefficient tables for the minimal polynomials of s(n)^2, called MPs2(n, x), for even and odd n have been given in A232631 and A232632, respectively. See these entries for details, and the link to the Q(2 cos(pi/n)) paper, Table 4, in A187360 for the power basis representation of the zeros of the minimal polynomial C(n, x) of rho(n).

The degree deg(n) of MPs2(n, x) is therefore delta(n/2) or delta(n) for n even or odd, respectively, where delta(n) = A055034(n). This means that deg(1) = deg(2) =1 and deg(n) = phi(n)/2 = A023022(n), n >= 3. deg(n) = A023022(n).

Especially MPs2(p, x) = Product_{j=0..(p-3)/2} (x - 2*(1 + cos(Pi*(2*j+1)/p))), for p an odd prime (A065091).

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called ``Polynomials associated with squared diagonals of regular polygons'', Nov 16 2013.

The length of row l of this table is delta(l) + 1 = A055034(l) + 1, l >= 1, that is: 2, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 5, ...

s(n):= 2*sin(Pi/n) is the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some length units). In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), the length ratio (smallest diagonal)/s(n) in the regular n-gon (n>=2). If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) + 2). Therefore s(2*l) is an integer in the algebraic number field Q(rho(l)).

Its (monic) minimal polynomial is obtained from the conjugates of rho(l), called rho(l;j), j = 1, 2, ..., delta(l), which are the zeros of the minimal polynomial of rho(l) = rho(l;1) of degree delta(l) = A055034(l), called C(l, x) in A187360. These conjugates are therefore rho(l;j) = 2*cos(Pi*rpnodd(l,j)/l) where rpnodd(l,j) is the j-th entry of the list rpnodd(l) of the odd numbers < l which are relatively prime to l (for example, rpnodd(9) = [1,5,7], and rpnodd(9,2) = 5). From this the conjugates of s(2*l)^2 become 2 - rho(l;j), and the minimal polynomial of s(2*l)^2 is MPs2(l, x) = product( x - (2- rho(l;j)), j=1..delta(l)) for l >=1. Because the zeros of C(l, x) are integers in the algebraic number field Q(rho(l)) written in its power basis (see table 4 of the link under A187360 to the Q(2 cos(Pi/n)) paper) one finds, after expansion and reducing powers of rho(l) modulo C(l, rho(l)), directly the integer coefficients appropriate for this (monic) minimal polynomial. Only the equation C(l, rho(l)) = 0 is needed, not the trigonometric version of rho(l) and its powers.

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called 'Polynomials associated with squared diagonals of regular polygons', Nov 16 2013.

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.