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.

4, -1, 9, -1, 29, 4, 99, 34, 349, 179, 1254, 824, 4559, 3574, 16704, 15004, 61549, 61709, 227799, 250229, 846254, 1004149, 3153984, 3997399, 11788879, 15812504, 44178624, 62229509, 165946124, 243873904, 624650004, 952400599, 2355748909, 3708579599

Offset: 0

Author

Greg Dresden, Feb 27 2019

Keywords

• sign
• easy

Comments

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}A187360,%20for%20n%20=%2015),%20appearing%20to%20the%20power%20n%20in%20the%20formula%20given%20above.%20-%20_Wolfdieter%20Lang">{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

Links

• Index entries for linear recurrences with constant coefficients, signature (-1,4,4,-1).

Crossrefs

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A324602.

Programs

Mathematica

Table[Sum[(2.0 Cos[k Pi/15])^n, {k, {1, 7, 11, 13}}] // Round, {n, 1, 30}]
LinearRecurrence[{-1,4,4,-1},{4,-1,9,-1},40] (* Harvey P. Dale, Jun 02 2024 *)

Formula

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

a(n) = -a(n-1) + 4*a(n-2) + 4*a(n-3) -a(n-4). - Greg Dresden, Feb 27 2019