A231185 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)). Coefficients of the third power.

1, 0, 4, 1, 14, 7, 48, 35, 165, 154, 572, 636, 2002, 2533, 7071, 9861, 25176, 37810, 90251, 143451, 325358, 540155, 1178291, 2022735, 4282811, 7543771, 15612092, 28048829, 57040186, 104050724, 208772476, 385320419, 765186422, 1425038684

Offset

0,3

Comments

This sequence gives the first differences of A231181.

The formula for rho(11)^n is (see A231182): rho(11)^n = A231182(n)*1 - A231183(n)*rho(11) - A231184(n-2)*rho(11)^2 + a(n-3)*rho(11)^3 + A231182(n+1)*rho(11)^4, n >= 0.

Links

Table of n, a(n) for n=0..33.

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

Formula

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

a(n) = b(n) - b(n-1) for n>=0, with b(n) = A231181(n) (first differences).

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, 4, 1, 14}, 40] (* Harvey P. Dale, Aug 03 2023 *)

Crossrefs

Cf. A231181, A231182, A231183, A231184.

Sequence in context: A327352 A050156 A191584 * A187055 A257501 A096644

Adjacent sequences: A231182 A231183 A231184 * A231186 A231187 A231188

Keyword

sign,easy

Author

Wolfdieter Lang, Nov 07 2013

Status

approved