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Array of coefficients of the minimal polynomial of (2*cos(Pi/15))^n, for n >= 1 (ascending powers).
0

%I #16 May 18 2019 03:51:57

%S 1,-4,-4,1,1,1,-24,26,-9,1,1,-109,-49,1,1,1,-524,246,-29,1,1,-2504,

%T -619,-4,1,1,-11979,2621,-99,1,1,-57299,-7774,-34,1,1,-274084,30126,

%U -349,1,1,-1311049,-97879,-179,1,1,-6271254,363131,-1254,1,1,-29997829,-1237504,-824,1

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

%C The length of each row is 5.

%C 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).

%F 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.

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

%e The rectangular array T(n, k) begins:

%e n\k 0 1 2 3 4

%e ---------------------------------

%e 1: 1 -4 -4 1 1

%e 2: 1 -24 26 -9 1

%e 3: 1 -109 -49 1 1

%e 4: 1 -524 246 -29 1

%e 5: 1 -2504 -619 -4 1

%e 6: 1 -11979 2621 -99 1

%e 7: 1 -57299 -7774 -34 1

%e ...

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

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

%K sign,tabf,easy

%O 1,2

%A _Greg Dresden_ and _Wolfdieter Lang_, May 02 2019