A232632 Coefficient table for the minimal polynomials of s(2*l+1)^2 = (2*sin(Pi/(2*l+1)))^2.

0, 1, -3, 1, 5, -5, 1, -7, 14, -7, 1, -3, 9, -6, 1, -11, 55, -77, 44, -11, 1, 13, -91, 182, -156, 65, -13, 1, 1, -8, 14, -7, 1, 17, -204, 714, -1122, 935, -442, 119, -17, 1, -19, 285, -1254, 2508, -2717, 1729, -665, 152, -19, 1, 1, -16, 60, -78, 44, -11, 1, -23, 506, -3289, 9867, -16445, 16744, -10948, 4692, -1311, 230, -23, 1

Offset: 0

Author

Wolfdieter Lang, Dec 18 2013

Keywords

• sign
• tabf
• easy

Comments

The length of row l is delta(2*l+1) + 1 = A055034(2*l+1) + 1, l >= 0.

See the comments on A232631 (even n case) for s(n) = 2*sin(Pi/n) and the minimal polynomial of s(n)^2. Here n = 2*l+1 and s(2*l+1)^2 = 4 - rho(2*l+1)^2 is an integer in the algebraic number field Q(rho(2*l+1)). The minimal polynomial of s(2*l+1)^2 is then MPs2(2*l+1, x) = product(x - 2*(1 + cos(Pi*rpnodd(2*l+1,j)/(2*l+1))), j=1..delta(2*l+1)), l >= 0, where rpnodd(2*l+1) is the list of positive odd numbers < 2*l+1 and relatively prime to 2*l+1. rpnodd(2*l+1,j) is the j-th member of this increasingly ordered list. Here the identity 4 - (2*cos(Pi*(2*k+1)/(2*l+1)))^2 = 2*(1 - cos(Pi*2*(2*k+1)/(2*l+1))) = 2*(1 - (- cos(Pi*(2*l+1 - 2*(2*k+1))/ (2*l+1)))) has been used, and for 2*k+1 < 2*l+1 and gcd(2*k+1, 2*l+1) = 1 this becomes the product given above because 1 = gcd(-(2*k+1), 2*l+1) = gcd(-2*(2*k+1), 2*l+1) = gcd(2*l+1, -2*(2*k+1) + (2*l+1)).

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.

Examples

			The table a(l,m) begins (n = 2*l+1):
------------------------------------------------------------------------------------------------------
n,   l\m     0      1      2       3        4       5        6       7     8      9    10  11 ...
1,   0:      0      1
3,   1:     -3      1
5,   2:      5     -5      1
7,   3:     -7     14     -7       1
9,   4:     -3      9     -6       1
11,  5:    -11     55    -77      44      -11       1
13,  6:     13    -91    182    -156       65     -13        1
15,  7:      1     -8     14      -7        1
17,  8:     17   -204    714   -1122      935    -442      119     -17      1
19,  9:    -19    285  -1254    2508    -2717    1729     -665     152    -19     1
21, 10:      1    -16     60     -78       44     -11        1
23, 11:    -23    506  -3289    9867   -16445   16744   -10948    4692  -1311   230   -23   1
25, 12:      5   -125    875   -2675     4300   -4005     2275    -800    170   -20     1
27, 13:     -3     81   -540    1386    -1782    1287     -546     135    -18     1
....
n=29, l=14:  29,-1015,10556,-51272,140998,-243542,281010,-224808,127281,-51359,14674, -2900,377,-29,1.
n=31, l=15: -31, 1240, -14756, 82212, -260338, 520676, -700910, 660858, -447051, 219604, -78430, 20150, -3627, 434, -31, 1.
...
The minimal polynomial of s(5)^2 = (2*sin(Pi/5))^2 = 4 - rho(5)^2
= 2*(1 - cos(Pi*2/5)) = 2*(1 + cos(Pi*3/5)),  approx. 1.381966, is MPs2(5, x) =  product(x - 2*(1 + cos(Pi*rpnodd(5,j)/5)), j=1..2) = (x - 2*(1 + cos(Pi/5))*(x - 2*(1 + cos(Pi*3/5)) = (x - (2 + phi)*(x - (2 + 1 - phi)) = x^2 - 5*x + (6 + phi - phi^2) = x^2 - 5*x +5, where phi = rho(5) is the golden section.
The row n=17 checks with WolframAlpha's MinimalPolynomial[(2*sin(Pi/17))^2 ,x] = 17-204 x+714 x^2-1122 x^3+935 x^4-442 x^5+119 x^6-17 x^7+x^8.
		

Crossrefs

Cf. A232631 (even n), A232633 (all n), A055034.

Programs

Mathematica

Flatten[ CoefficientList[ Table[ MinimalPolynomial[ (2*Sin[Pi/(2*l+1)])^2, x], {l, 0, 11}], x]] (* adapted from Jean-François Alcover, A187360 *) (* Wolfdieter Lang, Dec 23 2013 *)

Formula

a(l,m) = [x^m] MPs2(2*l+1, x), l >= 1, m = 0, 1, ...., delta(l), with the minimal polynomial MPs2(l, x) of (2*sin(Pi/(2*l+1)))^2, explained above in a comment.