A193640 Indices n such that Perrin(n) > r^n where r is the real root of the polynomial x^3-x-1.

0, 2, 3, 5, 8, 10, 13, 15, 16, 18, 20, 21, 23, 26, 28, 31, 33, 34, 36, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 62, 64, 65, 67, 69, 70, 72, 75, 77, 80, 82, 83, 85, 87, 88, 90, 93, 95, 96, 98, 100, 101, 103, 106, 108, 111, 113, 114, 116, 118, 119, 121, 124

Offset: 0

Author

Francesco Daddi, Aug 02 2011

Keywords

• nonn

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).

Examples

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

Crossrefs

Cf. A001608, A060006, A051017.

Programs

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 *)