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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.

%I A231185 #8 Aug 03 2023 19:31:27
%S A231185 1,0,4,1,14,7,48,35,165,154,572,636,2002,2533,7071,9861,25176,37810,
%T A231185 90251,143451,325358,540155,1178291,2022735,4282811,7543771,15612092,
%U A231185 28048829,57040186,104050724,208772476,385320419,765186422,1425038684
%N 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.
%C A231185 This sequence gives the first differences of A231181.
%C A231185 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.
%H A231185 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-3,1).
%F A231185 G.f.: (1 - x)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
%F A231185 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.
%F A231185 a(n) = b(n) - b(n-1) for n>=0, with b(n) = A231181(n)  (first differences).
%e A231185 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.
%t A231185 LinearRecurrence[{1,4,-3,-3,1},{1,0,4,1,14},40] (* _Harvey P. Dale_, Aug 03 2023 *)
%Y A231185 Cf. A231181, A231182, A231183, A231184.
%K A231185 sign,easy
%O A231185 0,3
%A A231185 _Wolfdieter Lang_, Nov 07 2013