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%I #16 Aug 03 2014 14:01:32
%S 1,4,6,7,9,11,12,14,17,19,22,24,25,27,29,30,32,35,37,38,40,42,43,45,
%T 48,50,53,55,56,58,60,61,63,66,68,71,73,74,76,78,79,81,84,86,89,91,92,
%U 94,97,99,102,104,105,107,109,110,112,115,117,120,122,123,125
%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). [From Francesco Daddi, Aug 02 2011]
%e For n=27 Perrin(27) = A001608(27) = 1983 < 1983.044... = r^27
%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, A051016.
%K nonn
%O 1,2
%A _Francesco Daddi_, Aug 01 2011
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