A205579 -id:A205579 - cp's OEIS frontend

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

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 22, 29, 38, 51, 67, 89, 119, 157, 209, 276, 366, 486, 643, 853, 1130, 1496, 1983, 2626, 3480, 4610, 6106, 8090, 10716, 14196, 18807, 24913, 33004, 43721, 57917, 76725, 101638, 134643, 178364, 236281, 313007, 414645

Offset: 0

Roger L. Bagula, Mar 31 2006

nonn

Andrei Vieru, Pisot Numbers and Primes, arXiv:1205.1054 [math.NT], Apr 04 2012.

Cf. A060006 (decimal expansion of r=1.32471795724475...).
Cf. A051016, A051017.
Cf. A205579 (definition using round() instead of floor()).

r = Solve[x^3 - x - 1 == 0, x][[1,1,2]]; Table[Floor[r^n], {n, 0, 50}] (* T. D. Noe, Jan 30 2012 *)

default(realprecision,110);
default(format,"g.15");
r=real(polroots(x^3-x-1)[1])
v=vector(66, n, floor(r^(n-1)) )  /* Joerg Arndt, Jan 29 2012 */

a(n) = floor(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n). - Jwalin Bhatt, May 06 2025

Completely edited by Joerg Arndt, Jan 29 2012

A382641 a(n) = round(c^n), where c is the supergolden ratio A092526.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907, 13749853

Offset: 0

Jwalin Bhatt, Apr 01 2025

Wikipedia, Supergolden ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A001609, A092526, A169985, A205579.

r = Root[x^3-x^2-1, 1]; Table[Round[r^i], {i,0,120 }]
CoefficientList[Series[(1+x^2+x^4-x^8)/(1-x-x^3), {x,0,120}], x]

G.f.: (1 + x^2 + x^4 - x^8)/(1 - x - x^3).
a(n) = a(n-1) + a(n-3) for n>=9.
a(n) = round(((2/3)*cos((1/3)*arccos(29/2))+1/3)^n) = round(A092526^n).
a(n) = A001609(n) for n>=6.

cp's OEIS Frontend

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

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