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.

1, 0, 0, 0, 0, 1, 1, 5, 6, 20, 27, 75, 110, 275, 429, 1001, 1637, 3639, 6172, 13243, 23104, 48280, 86090, 176341, 319792, 645150, 1185305, 2363596, 4386331, 8669142, 16212913, 31825005, 59873834, 116914020, 220964744, 429737220, 815057639

Offset: 0

Author

Wolfdieter Lang, Nov 05 2013

Keywords

• nonn
• easy

Comments

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.

Examples

			rho(11)^4 = 0*1 - 0*rho(11) - 0*rho(11)^2 + 0*rho(11)^3 + 1*rho(11)^4 (trivial).
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.
		

Links

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

Crossrefs

Cf. A231181, A231183, A231184, A231185.

Formula

G.f.: (1-x-x^2)*(1-3*x^2)/(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>= 5, with a(0)=1, a(1)=a(2)=a(3)=a(4)=0.

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).