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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.
4
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.
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
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Nov 07 2013
STATUS
approved