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

0,3

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).

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)

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 *)

Prog

(PARI)
default(realprecision, 110);
default(format, "g.15");
r=real(polroots(x^3-x-1)[1])
v=vector(66, n, round(r^(n-1)) )
(PARI) Vec((1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3)+O(x^66))

Crossrefs

Cf. A112639 (definition using floor() instead of round()).

Cf. A060006 (decimal expansion of r=1.32471795724475...).

Cf. A051016, A051017, A169985.

Sequence in context: A064324 A173090 A032277 * A089047 A133498 A282949

Adjacent sequences: A205576 A205577 A205578 * A205580 A205581 A205582

Keyword

nonn,easy

Author

Joerg Arndt, Jan 29 2012

Status

approved