A231188 Coefficient table for the minimal polynomials of 2*sin(2*Pi/n). Rising powers of x.

0, 1, 0, 1, -3, 0, 1, -2, 1, 5, 0, -5, 0, 1, -3, 0, 1, -7, 0, 14, 0, -7, 0, 1, -2, 0, 1, -3, 0, 9, 0, -6, 0, 1, 5, 0, -5, 0, 1, -11, 0, 55, 0, -77, 0, 44, 0, -11, 0, 1, -1, 1, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, -7, 0, 14, 0, -7, 0, 1, 1, 0, -8, 0, 14, 0, -7, 0, 1, 2, 0, -4, 0, 1, 17, 0, -204, 0, 714, 0, -1122, 0, 935, 0, -442, 0, 119, 0, -17, 0, 1, -3, 0, 9, 0, -6, 0, 1

Offset: 1

Author

Wolfdieter Lang, Nov 29 2013

Keywords

• sign
• tabf

Comments

The length of row n is deg(n) + 1 = A093819(n) + 1, that is 2, 2, 3, 2, 5, 3, 7, 3, 7, 5, 11, 2, 13, 7, 9, 5, 17,...

See A181871 for the coefficient table for the integer but non-monic minimal polynomials of sin(2*Pi/n), n>=1, called there pi(n, x). The present minimal polynomials of 2*sin(2*Pi/n) are integer and monic, and they are given by

MP2sin2(n, x) = pi(n, x/2).

Examples

			The table a(n,m) starts:
---------------------------------------------------------------------------------
n\m   0   1    2  3    4  5     6  7    8  9    10  11   12  13   14 15 16 ...
1:    0   1
2:    0   1
3:   -3   0    1
4:   -2   1
5:    5   0   -5  0    1
6:   -3   0    1
7:   -7   0   14  0   -7  0     1
8:   -2   0    1
9:   -3   0    9  0   -6  0     1
10:   5   0   -5  0    1
11: -11   0   55  0  -77  0    44  0  -11  0     1
12:  -1   1
13:  13   0  -91  0  182  0  -156  0   65  0   -13   0    1
14:  -7   0   14  0   -7  0     1
15:   1   0   -8  0   14  0    -7  0    1
16:   2   0   -4  0    1
17:  17   0 -204  0  714  0 -1122  0  935  0  -442   0  119   0  -17  0  1
...
		

Crossrefs

Cf. A093819, A181871, A181872/A181872, A232624.

Formula

a(n,m) = [x^m] MP2sin2(n, x), n>=1, m = 0, 1, ..., A093819(n), with the minimal polynomials of 2*sin(2*Pi/n), given above in a comment in terms of the ones for sin(2*Pi/n).