A231183 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 first power.

0, -1, 0, 0, 0, 3, 2, 14, 13, 54, 61, 198, 255, 715, 1012, 2574, 3910, 9280, 14877, 33557, 56069, 121736, 209990, 442933, 783035, 1615658, 2910765, 5905483, 10795397, 21621095, 39969597, 79262102, 147796497, 290868226, 545980212, 1068246916

Offset

0,6

Comments

The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - a(n)*rho(11) - A231184(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..35.

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

Formula

G.f.: x*(-1 + x + 4*x^2 -3*x^3)/(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 >= 5, with a(0)=0, a(1)=-1, a(2)=a(3)=a(4)=0.

a(n) = -b(n-1) + b(n-2) + 4*b(n-3) - 3*b(n-4) for 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.

Crossrefs

Cf. A231181, A231182, A231184, A231185.

Sequence in context: A361456 A324012 A355259 * A324661 A163355 A214885

Adjacent sequences: A231180 A231181 A231182 * A231184 A231185 A231186

Keyword

sign,easy

Author

Wolfdieter Lang, Nov 07 2013

Status

approved