cp's OEIS Frontend

A231182 Coefficients for 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 for the zeroth and fourth powers.

%I A231182 #10 Feb 10 2024 09:22:45
%S A231182 1,0,0,0,0,1,1,5,6,20,27,75,110,275,429,1001,1637,3639,6172,13243,
%T A231182 23104,48280,86090,176341,319792,645150,1185305,2363596,4386331,
%U A231182 8669142,16212913,31825005,59873834,116914020,220964744,429737220,815057639
%N A231182 Coefficients for 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 for the zeroth and fourth powers.
%C A231182 The formula for rho(11)^n, with rho(11) = 2*cos(Pi/11) (the length ratio (smallest diagonal)/side in the regular 11-gon) written in the power basis of the number field  Q(rho(11)) is: rho(11)^n = a(n)*1 - A231183(n)*rho(11) - A231184(n-2)* rho(11)^2 + A231185(n-3)*rho(11)^3 + a(n+1)*rho(11)^4, n >= 0.
%H A231182 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-3,-3,1).
%F A231182 G.f.: (1-x-x^2)*(1-3*x^2)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
%F A231182 a(n) = a(n-1) + 4*a(n-2) - 3*a(n-3) - 3*a(n-4) + a(n-5) for n>= 5, with a(0)=1, a(1)=a(2)=a(3)=a(4)=0.
%F A231182 a(n) = b(n) - b(n-1) - 4*b(n-2) + 3*b(n-3) + 3*b(n-4) for n>=0, with b(n) = A231181(n).
%e A231182 rho(11)^4 = 0*1 - 0*rho(11) - 0*rho(11)^2 + 0*rho(11)^3 + 1*rho(11)^4 (trivial).
%e A231182 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.
%Y A231182 Cf. A231181, A231183, A231184, A231185.
%K A231182 nonn,easy
%O A231182 0,8
%A A231182 _Wolfdieter Lang_, Nov 05 2013