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.

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

Offset: 0

Author

Wolfdieter Lang, Nov 04 2013

Keywords

• nonn
• easy

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).

Examples

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

Links

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

Crossrefs

Cf. A038342, A069006, A230081.

Programs

Mathematica

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

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.