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%I #20 Aug 03 2014 14:01:32
%S 1,2,4,7,9,12,14,15,17,19,20,22,25,27,30,32,33,35,37,38,40,43,45,48,
%T 50,51,53,56,58,61,63,64,66,68,69,71,74,76,79,81,82,84,86,87,89,92,94,
%U 97,99,100,102,104,105,107,110,112,113,115,117,118,120,123
%N Indices n such that Padovan(n) < r^n/(2*r+3) where r is the real root of the polynomial x^3-x-1.
%C R is the so-called plastic number (A060006). Padovan(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r (real), s, t are the three roots of x^3-x-1. Also Padovan(n) is asymptotic to r^n / (2*r+3).
%e For n=25, Padovan(25) = A000931(25) = 200 < 200.023... = r^25/(2*r+3).
%t lim=200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, lim}]; p = Rest[CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]]; Select[Range[lim], p[[#]] <= powers[[#]] &] (* _T. D. Noe_, Aug 01 2011 *)
%Y Cf. A000931, A060006.
%K nonn
%O 1,2
%A _Francesco Daddi_, Jul 31 2011
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