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A205579 a(n) = round(r^n) where r is the smallest Pisot number (real root r=1.3247179.. of x^3-x-1).

%I A205579 #35 Apr 01 2025 23:10:21
%S A205579 1,1,2,2,3,4,5,7,9,13,17,22,29,39,51,68,90,119,158,209,277,367,486,
%T A205579 644,853,1130,1497,1983,2627,3480,4610,6107,8090,10717,14197,18807,
%U A205579 24914,33004,43721,57918,76725,101639,134643,178364,236282,313007,414646,549289,727653,963935,1276942,1691588,2240877,2968530,3932465
%N A205579 a(n) = round(r^n) where r is the smallest Pisot number (real root r=1.3247179.. of x^3-x-1).
%H A205579 Vincenzo Librandi, <a href="/A205579/b205579.txt">Table of n, a(n) for n = 0..1000</a>
%H A205579 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PisotNumber.html">Pisot Number</a>.
%H A205579 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1).
%F A205579 G.f.: (1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3).
%F A205579 From _Jwalin Bhatt_, Mar 26 2025: (Start)
%F A205579 a(n) = round(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n).
%F A205579 a(n) = a(n-2) + a(n-3) for n>=13. (End)
%t A205579 CoefficientList[Series[(1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3),{x,0,100}],x] (* _Vincenzo Librandi_, Aug 19 2012 *)
%t A205579 r = Root[x^3-x-1, 1]; Table[Round[r^i], {i,0,100 }] (* _Jwalin Bhatt_, Mar 27 2025 *)
%o A205579 (PARI)
%o A205579 default(realprecision, 110);
%o A205579 default(format, "g.15");
%o A205579 r=real(polroots(x^3-x-1)[1])
%o A205579 v=vector(66, n, round(r^(n-1)) )
%o A205579 (PARI) Vec((1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3)+O(x^66))
%Y A205579 Cf. A112639 (definition using floor() instead of round()).
%Y A205579 Cf. A060006 (decimal expansion of r=1.32471795724475...).
%Y A205579 Cf. A051016, A051017, A169985.
%K A205579 nonn,easy
%O A205579 0,3
%A A205579 _Joerg Arndt_, Jan 29 2012