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A230081

Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in the power basis of the degree 5 number field Q(rho(11)). Negative of the coefficients of the second power.

0, 4, 13, 50, 173, 613, 2149, 7557, 26543, 93264, 327657, 1151183, 4044478, 14209634, 49923211, 175397097, 616229093, 2165020570, 7606447024, 26724012524, 93890464368, 329868851103, 1158940469863, 4071748539926, 14305425173111

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OFFSET

0,2

COMMENTS

The formula for the nonpositive powers of rho(11) := 2*cos(Pi/11) (the length ratio (smallest diagonal/side) in the regular 11-gon), when written in the power basis of the degree 5 algebraic number field Q(rho(11)) is: 1/rho(11)^n = A038342(n)*1 + A230080*rho(11) - a(n)*rho(11)^2 - A069006(n-1)*rho(11)^3 + A038342(n-1)*rho(11)^4, n >= 0, with A069006(-1) = 0 = A038342(-1).

LINKS

Table of n, a(n) for n=0..24.

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

FORMULA

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

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

EXAMPLE

1/rho(11)^4 = 146*1 + 73*rho(11) - 173*rho(11)^2 - 29*rho(11)^3 + 41*rho(11)^4 (approximately 0.07374164519).

CROSSREFS

Cf. A038342, A069006, A230080.

Sequence in context: A149454 A101125 A284336 * A364907 A299699 A056275

Adjacent sequences: A230078 A230079 A230080 * A230082 A230083 A230084