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%I #32 May 09 2019 10:04:31
%S 4,24,109,524,2504,11979,57299,274084,1311049,6271254,29997829,
%T 143491199,686373809,3283190949,15704770004,75121978804,359337430474,
%U 1718849676159,8221921677724,39328626006254,188124003629279,899869747188249,4304424455586134
%N a(n) = (2*cos(Pi/15))^(-n) + (2*cos(7*Pi/15))^(-n) + (2*cos(11*Pi/15))^(-n) + (2*cos(13*Pi/15))^(-n), for n >= 1.
%C -a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).
%C a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}_{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - _Wolfdieter Lang_, May 08 2019
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-1,-1).
%F a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).
%F G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).
%F a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.
%t Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k,{1,7,11,13}}],50]],{n,1,30}]
%Y Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.
%K nonn,easy
%O 1,1
%A _Greg Dresden_, Feb 28 2019
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