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A306603 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.

%I A306603 #32 Jun 02 2024 12:16:11
%S A306603 4,-1,9,-1,29,4,99,34,349,179,1254,824,4559,3574,16704,15004,61549,
%T A306603 61709,227799,250229,846254,1004149,3153984,3997399,11788879,15812504,
%U A306603 44178624,62229509,165946124,243873904,624650004,952400599,2355748909,3708579599
%N A306603 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.
%C A306603 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 = -1, e_2 = -4, e_3 = -4 and e_4 = 1. The arguments are e_j(x_1, x_2, x_3, x_4), for j = 1..4, with the zeros {x_i}_{i=1..4} of the minimal polynomial of 2*cos(Pi/15) (see A187360, for n = 15), appearing to the power n in the formula given above. - _Wolfdieter Lang_, May 08 2019
%H A306603 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,4,4,-1).
%F A306603 G.f.: (4*x^3+8*x^2-3*x-4)/(-x^4+4*x^3+4*x^2-x-1). - _Alois P. Heinz_, Feb 27 2019
%F A306603 a(n) = -a(n-1) + 4*a(n-2) + 4*a(n-3) -a(n-4). - _Greg Dresden_, Feb 27 2019
%t A306603 Table[Sum[(2.0 Cos[k Pi/15])^n, {k, {1, 7, 11, 13}}] // Round, {n, 1, 30}]
%t A306603 LinearRecurrence[{-1,4,4,-1},{4,-1,9,-1},40] (* _Harvey P. Dale_, Jun 02 2024 *)
%Y A306603 Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A324602.
%K A306603 sign,easy
%O A306603 0,1
%A A306603 _Greg Dresden_, Feb 27 2019