|
|
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.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,8
|
|
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.
|
|
LINKS
|
|
|
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).
|
|
EXAMPLE
|
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.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|