A205579 a(n) = round(r^n) where r is the smallest Pisot number (real root r=1.3247179.. of x^3-x-1).
1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367, 486, 644, 853, 1130, 1497, 1983, 2627, 3480, 4610, 6107, 8090, 10717, 14197, 18807, 24914, 33004, 43721, 57918, 76725, 101639, 134643, 178364, 236282, 313007, 414646, 549289, 727653, 963935, 1276942, 1691588, 2240877, 2968530, 3932465
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Pisot Number.
- Index entries for linear recurrences with constant coefficients, signature (0,1,1).
Crossrefs
Programs
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Mathematica
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 *) r = Root[x^3-x-1, 1]; Table[Round[r^i], {i,0,100 }] (* Jwalin Bhatt, Mar 27 2025 *) -
PARI
default(realprecision, 110); default(format, "g.15"); r=real(polroots(x^3-x-1)[1]) v=vector(66, n, round(r^(n-1)) )
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PARI
Vec((1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3)+O(x^66))
Formula
G.f.: (1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3).
From Jwalin Bhatt, Mar 26 2025: (Start)
a(n) = round(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n).
a(n) = a(n-2) + a(n-3) for n>=13. (End)