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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.
4
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
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).
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 04 2013
STATUS
approved