A232629 Coefficients of the algebraic number 2*sin(4*Pi/n) in the power basis of Q(2*cos(Pi/q(2,n))), with q(2,n) = A232625(n), n >= 1.

0, 0, 0, -1, 0, 0, -3, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 0, 0, -7, 0, 14, 0, -7, 0, 1, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, -5, 0, 1, 0, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1

Offset: 1

Author

Wolfdieter Lang, Dec 17 2013

Keywords

• sign
• tabf

Comments

The length of row n is A232626(n).

In a regular n-gon, n>=2, inscribed in a circle of radius R (in some length units), 2*sin(4*Pi/n) = (S(n)/R)*(D(1,n)/S(n)) = D(1,n)/R, with the side length S(n) and the length of the first (smallest) diagonal D(1,n). For n=2 there is no such diagonal, and one can put D(1,2) = 0. Obviously, D(1,2*m) = S(m), m >= 2.

See a comment on A231190 regarding the pair of sequences p(k,n) and q(k,n), n >= 1, k >= 1. Here k=2 with A231190 and A232625.

See also the k=1 analog A231189 of the present table.

The relevant identity is here 2*sin(Pi*4/n) = 2*cos(Pi*abs(n-8)/(2*n)) = 2*cos(Pi*p(2,n)/q(2,n)). This is R(p(2,n), x) (mod C(q(2,n), x)), with x = 2*cos(Pi/q(2,n)) =: rho(q(2,n)). with the coefficient tables for the polynomials R and C given in A127672 and A187360, respectively. This gives the power base coefficients of 2*sin(Pi*4/n) in the algebraic number field Q(rho(q(2,n))) of degree delta(q(2,n)), with delta(n) = A055034(n), shown in A232626.

If the degree p(2,n) of R(p(2,n), x) is smaller than the degree A232626(n) of C(q(2,n), x) then 2*sin(Pi*4/n) = R(p(2,n), x). Otherwise the (mod C(q(2,n), x)) congruence is needed. This happens for n = 1, 2, 3, 4, 21, 24, 27, 30,...

The power basis of Q(rho(q(2,n))) is <1, rho(q(2,n)), ..., rho(q(2,n))^(delta(q(2,n))-1)>. Therefore the length of row n of this table is delta(q(2,n)) = A232626(n).

The coefficient table for the minimal polynomial of 2*sin(Pi*4/n) is given in A232630.

Examples

			The table a(n,m) begins (the trailing zeros are needed to have the correct degree A232626(n) in Q(rho(q(2,n))))
-----------------------------------------------------------------
n\m 0   1  2   3  4   5  6   7  8   9 10 11 12 13 14 15 16 17 ...
1:  0
2:  0
3:  0  -1
4:  0
5:  0  -3  0   1
6:  0   1
7:  0   1  0   0  0   0
8:  2
9:  0   1  0   0  0   0
10: 0   1  0   0
11: 0  -3  0   1  0   0  0   0  0   0
12: 0   1
13: 0   5  0  -5  0   1  0   0  0   0  0  0
14: 0  -3  0   1  0   0
15: 0  -7  0  14  0  -7  0   1
16: 0   1
17: 0   9  0 -30  0  27  0  -9  0   1  0  0  0  0  0  0
18: 0   5  0  -5  0   1
19: 0 -11  0  55  0 -77  0  44  0 -11  0  1  0  0  0  0  0  0
20: 0  -3  0   1
...
n=1: 2*sin(Pi*4/1) = 0. R(p(2,1), x) = R(7, x) = -7*x + 14*x^3 -7*x^5 + x^7. C(q(2,1), x) = C(2, x) = x, hence R(7, x) (mod C(2, x)) == 0, and
  with A232626(1) = 1, a(1,0) = 0.n=7: p(2,7) = A231190(7) = 1, q(2,7) = A232625(7) = 14. 2*sin(Pi*4/7) = R(1, x) = x = rho(14) := 2*cos(Pi/14). C(14, x) of degree 6 does not apply here. A232626(7) = 6, hence the row n=7 is  0  1  0  0  0  0.
n=9: p(2,9) = 1, q(2,9) = 18. 2*sin(Pi*4/9) = R(1, x) = x = rho(18) = 2*cos(Pi/18). C(18, x) with degree 6 is not needed here. A232626(9) = 6, hence row n=9 is also 0  1  0  0  0  0.
n=8: this row with entry 2 coincides with row n=4 of A231189.
n=17: row length A232626(17) = 16; p(2,17) = 9; C(34, x) has degree 16, therefore the R(9, x) coefficients produce here the first 10 entries for row n=17: 0  9  0 -30  0  27  0 -9  0  1, followed by 6 zeros, and 2*sin(Pi*4/17) = 9*rho(34) - 30*rho(34)^3 + 27*rho(34)^5 - 9*rho(34)^7 + 1*rho(34)^9, with rho(34) = 2*cos(Pi/34).
		

Crossrefs

Cf. A231190 (p), A232625 (q), A127672 (R), A187360 (C), A232626 (degree), A231189 (k=1 case), A232630 (minimal polynomials).

Formula

a(n,m) = [x^m] (R(p(2,n), x) (mod C(q(2,n), x)), n >= 1, m = 0, 1, ..., A232626(n) - 1, where the C and R polynomials are found in A187360 and A127672, respectively. p(2,n) = A231190(n) and q(2,n) = A232625(n). Powers of x = rho(q(2,n)) := 2*cos(Pi/q(2,n)) appear in the table in increasing order.

a(2*l,m) = A231189(l,m), l >= 1, m = 0, 1, ..., (A093819(n)-1).