OFFSET
0,2
COMMENTS
r is the so-called plastic number (A060006).
Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.
Also Perrin(n) is asymptotic to r^n.
To calculate r^n (for n>2) we can observe that: r^n=s(n)*r^2+t(n)*r+u(n) where s(3)=0, t(3)=1, u(3)=1; s(n+1)=t(n), t(n+1)=s(n)+u(n), u(n+1)=s(n).
EXAMPLE
For n=20 Perrin(20) = A001608(20) = 277 > 276.992... = r^20
MATHEMATICA
lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n], {n, 0, lim}]; p = CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim + 1], p[[#]] > powers[[#]] &] - 1 (* T. D. Noe, Aug 02 2011 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Francesco Daddi, Aug 02 2011
STATUS
approved