A307886 Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).

1, -4, -4, 1, 1, 1, -24, 26, -9, 1, 1, -109, -49, 1, 1, 1, -524, 246, -29, 1, 1, -2504, -619, -4, 1, 1, -11979, 2621, -99, 1, 1, -57299, -7774, -34, 1, 1, -274084, 30126, -349, 1, 1, -1311049, -97879, -179, 1, 1, -6271254, 363131, -1254, 1, 1, -29997829, -1237504, -824, 1

Offset: 1

Author

Greg Dresden and Wolfdieter Lang, May 02 2019

Keywords

• sign
• tabf
• easy

Comments

The length of each row is 5.

The minimal polynomial of (2*cos(Pi/15))^n, for n >= 1, is C(15, n, x) = Product_{j=0..3} (x - (x_j)^n) = Sum_{k=0} T(n, k) x^k, where x_0 = 2*cos(Pi/15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15), and x_3 = 2*cos(13*Pi/15) are the zeros of C(15, 1, x) with coefficients given in A187360 (row n=15).

Examples

			The rectangular array T(n, k) begins:
n\k 0      1      2      3      4
---------------------------------
1:  1     -4     -4      1      1
2:  1    -24     26     -9      1
3:  1   -109    -49      1      1
4:  1   -524    246    -29      1
5:  1  -2504   -619     -4      1
6:  1 -11979   2621    -99      1
7:  1 -57299  -7774    -34      1
...
		

Crossrefs

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A306603, A306610, A306611.

Programs

Mathematica

Flatten[Table[CoefficientList[MinimalPolynomial[(2*Cos[\[Pi]/15])^n, x], x], {n, 1, 15}]]

Formula

T(n,k) = the coefficient of x^k in C(15, n, x), n >= 1, k=0,1,2,3,4, with C(15, n, k) the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 as defined above.

T(n, 0) = T(n, 4) = 1. T(n, 1) = -A306610(n), T(n, 2) = A306611(n), T(n, 3) = -A306603(n), n >= 1.