A193627 Indices n such that Perrin(n) < r^n where r is the real root of the polynomial x^3-x-1.

1, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 24, 25, 27, 29, 30, 32, 35, 37, 38, 40, 42, 43, 45, 48, 50, 53, 55, 56, 58, 60, 61, 63, 66, 68, 71, 73, 74, 76, 78, 79, 81, 84, 86, 89, 91, 92, 94, 97, 99, 102, 104, 105, 107, 109, 110, 112, 115, 117, 120, 122, 123, 125

Offset: 1

Author

Francesco Daddi, Aug 01 2011

Keywords

• nonn
• changed

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). - Francesco Daddi, Aug 02 2011

Examples

			For n=27 Perrin(27) = A001608(27) = 1983 < 1983.044... = r^27
		

Crossrefs

Cf. A001608, A060006, A051016.

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 *)