A231184 Coefficients of the nonnegative powers of rho(11) = 2*cos(Pi/11) when written in the power basis of the degree 5 number field Q(rho(11)). Negative of the coefficients of the second power.

-1, 0, 0, 3, 6, 17, 32, 73, 135, 286, 528, 1080, 2002, 4015, 7485, 14827, 27796, 54606, 102869, 200909, 380006, 739013, 1402309, 2718485, 5171573, 10001553, 19064476, 36802823, 70259834, 135444612, 258883604, 498538557, 953762458

Offset

0,4

Comments

The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - A231183(n)*rho(11) - a(n-2)*rho(11)^2 + A231185(n-3)*rho(11)^3 + A231182(n+1)*rho(11)^4, n >= 0.

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-3,1).

Formula

G.f.: (-1 + x + 4*x^2)/(1-x-4*x^2+3*x^3+3*x^4-x^5).

a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n >= 3, with a(-2)=a(-1)=0 , a(0)=-1, a(1)=a(2)=0.

a(n) = -b(n) + b(n-1) + 4*b(n-2), n>=0, with b(n) = A231181(n).

Example

rho(11)^5 = 1*1 - 3*rho(11) - 3*rho(11)^2 + 4*rho(11)^3 + 1*rho(11)^4. Approximately 26.02309649, with rho(11) approximately 1.918985947.

Mathematica

LinearRecurrence[{1, 4, -3, -3, 1}, {-1, 0, 0, 3, 6}, 40] (* Harvey P. Dale, Apr 26 2019 *)

Crossrefs

Cf. A231181, A231182, A231183, A231185.

Sequence in context: A327068 A307604 A049943 * A291227 A027415 A280088

Adjacent sequences: A231181 A231182 A231183 * A231185 A231186 A231187

Keyword

sign,easy

Author

Wolfdieter Lang, Nov 07 2013

Status

approved