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A306611 The middle coefficient in the minimal polynomial for (2*cos(Pi/15))^n. 2
-4, 26, -49, 246, -619, 2621, -7774, 30126, -97879, 363131, -1237504, 4497801, -15702574, 56538746, -199764994, 716265246, -2545683874, 9110943101, -32474838004, 116135818131, -414537600379, 1481979727826, -5293483738474, 18921861083121, -67610126265619, 241664630238746 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Wolfdieter Lang, May 01 2019: (Start)

rho(15) = 2*cos(Pi/15) = 2*A019887 gives the length ratio of the smallest diagonal and the side of a regular 15-gon. The minimal polynomial of rho(15) is C(n, x) = x^4 + x^3 - 4*x^2 - 4*x + 1, with zeros x_0 = rho(15), x_1 = 2*cos(7*Pi/15), x_2 = 2*cos(11*Pi/15) and x_3 = 2*cos(13*Pi/15). See A187360, also for a W. Lang link.

The minimal polynomial of rho(1)^n, for n >= 1, considered here, is C(15,n,x) = Product_{j=0..3} (x - x_j^n) = x^4 - A_1(n)x^3 + A_2(n)*x^2 - A_3(n)*x + A_4(n). The coefficients are the elementary symmetric functions A_j(n) = sigma_j((x_0)^n, (x_1)^n, (x_2)^n, (x_3)^n), for j = 1, 2, 3, and  A_4(n) = (A_4(1))^n = 1. A_1(n) =  A306603(n), A_2(n) =  a(n), and A_3(n) = A306610(n), for n >= 1.

Thanks to Greg Dresden for sending me a proof that C(15,n,x) has integer coefficients and does not factor over the rationals for n >= 1. (End)

LINKS

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (-4,5,25,5,-4,-1).

FORMULA

a(n) = -4*a(n-1) + 5*a(n-2) + 25*a(n-3) + 5*a(n-4) - 4*a(n-5) - a(n-6).

G.f.: -x*(4 - 10*x - 75*x^2 - 20*x^3 + 20*x^4 + 6*x^5) / ((1 + 3*x + x^2)*(1 + x - 9*x^2 + x^3 + x^4)). - Colin Barker, Feb 28 2019

MATHEMATICA

Table[Coefficient[MinimalPolynomial[(2Cos[Pi/15])^n, x], x, 2], {n, 1, 40}]

PROG

(PARI) Vec(-x*(4 - 10*x - 75*x^2 - 20*x^3 + 20*x^4 + 6*x^5) / ((1 + 3*x + x^2)*(1 + x - 9*x^2 + x^3 + x^4)) + O(x^30)) \\ Colin Barker, Feb 28 2019

CROSSREFS

Cf. A306603 (which gives the negative coefficient of x^3 in minimal polynomial for (2 cos(Pi/15))^n) and A306610 (likewise for the coefficient of x).

Cf. A019887 (cos(Pi/15), A187360.

Sequence in context: A059178 A056193 A196672 * A102203 A219668 A273981

Adjacent sequences:  A306608 A306609 A306610 * A306612 A306613 A306614

KEYWORD

sign,easy

AUTHOR

Greg Dresden, Feb 28 2019

STATUS

approved

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Last modified August 20 06:04 EDT 2019. Contains 326139 sequences. (Running on oeis4.)