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A230080 Sequence needed for the nonpositive powers of rho(11) = 2*cos(Pi/11) in terms of the power basis of the degree 5 number field Q(rho(11)). Coefficients of the first power. 4
0, 3, 5, 23, 73, 265, 920, 3245, 11385, 40018, 140574, 493911, 1735243, 6096533, 21419128, 75252674, 264387942, 928884046, 3263482673, 11465714843, 40282921096, 141527481021, 497233748352, 1746949771565, 6137623429414 (list; graph; refs; listen; history; text; internal format)
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 + a(n)*rho(11) - A230081(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*(3 - 4*x - x^2 + x^3)/(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)=a(-3)=a(-2)=0, a(-1)=1.

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).

MATHEMATICA

LinearRecurrence[{3, 3, -4, -1, 1}, {0, 3, 5, 23, 73}, 30] (* Harvey P. Dale, May 19 2017 *)

CROSSREFS

Cf. A038342, A069006, A230081.

Sequence in context: A027753 A066411 A153410 * A155778 A209028 A178068

Adjacent sequences:  A230077 A230078 A230079 * A230081 A230082 A230083

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Nov 04 2013

STATUS

approved

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Last modified April 20 20:26 EDT 2021. Contains 343137 sequences. (Running on oeis4.)