A193640 - OEIS

%I #14 Aug 10 2014 13:44:41

%S 0,2,3,5,8,10,13,15,16,18,20,21,23,26,28,31,33,34,36,39,41,44,46,47,

%T 49,51,52,54,57,59,62,64,65,67,69,70,72,75,77,80,82,83,85,87,88,90,93,

%U 95,96,98,100,101,103,106,108,111,113,114,116,118,119,121,124

%N Indices n such that Perrin(n) > r^n where r is the real root of the polynomial x^3-x-1.

%C r is the so-called plastic number (A060006).

%C Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.

%C Also Perrin(n) is asymptotic to r^n.

%C 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).

%e For n=20 Perrin(20) = A001608(20) = 277 > 276.992... = r^20

%t 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 *)

%Y Cf. A001608, A060006, A051017.

%K nonn

%O 0,2

%A _Francesco Daddi_, Aug 02 2011