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A112639 a(n) = floor(r^n) where r is the smallest Pisot number (real root r=1.3247179... of x^3-x-1).

%I A112639 #36 May 06 2025 17:10:35
%S A112639 1,1,1,2,3,4,5,7,9,12,16,22,29,38,51,67,89,119,157,209,276,366,486,
%T A112639 643,853,1130,1496,1983,2626,3480,4610,6106,8090,10716,14196,18807,
%U A112639 24913,33004,43721,57917,76725,101638,134643,178364,236281,313007,414645
%N A112639 a(n) = floor(r^n) where r is the smallest Pisot number (real root r=1.3247179... of x^3-x-1).
%H A112639 Andrei Vieru, <a href="http://arxiv.org/abs/1205.1054">Pisot Numbers and Primes</a>, arXiv:1205.1054 [math.NT], Apr 04 2012.
%F A112639 a(n) = floor(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n). - _Jwalin Bhatt_, May 06 2025
%t A112639 r = Solve[x^3 - x - 1 == 0, x][[1,1,2]]; Table[Floor[r^n], {n, 0, 50}] (* _T. D. Noe_, Jan 30 2012 *)
%o A112639 (PARI)
%o A112639 default(realprecision,110);
%o A112639 default(format,"g.15");
%o A112639 r=real(polroots(x^3-x-1)[1])
%o A112639 v=vector(66, n, floor(r^(n-1)) )  /* _Joerg Arndt_, Jan 29 2012 */
%Y A112639 Cf. A060006 (decimal expansion of r=1.32471795724475...).
%Y A112639 Cf. A051016, A051017.
%Y A112639 Cf. A205579 (definition using round() instead of floor()).
%K A112639 nonn
%O A112639 0,4
%A A112639 _Roger L. Bagula_, Mar 31 2006
%E A112639 Completely edited by _Joerg Arndt_, Jan 29 2012