A018919 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(3,9).

3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 123111, 354484, 1020696, 2938977, 8462447, 24366645, 70160958, 202020427, 581694636, 1674922950, 4822748423, 13886550633, 39984728949, 115131438424, 331507764639, 954538564968

Offset

0,1

Comments

Let M denotes the 4 X 4 matrix = row by row (1,1,1,1)(1,1,1,0)(1,1,0,0)(1,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n))=M^n*A where A is the vector (1,1,1,1) then a(n)=y(n+1). - Benoit Cloitre, Apr 02 2002

Not to be confused with the Pisot T(3,9) sequence, which is A000244. - R. J. Mathar, Feb 13 2016

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.

Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Index entries for Pisot sequences

Formula

For n>1, a(n) = ceiling(a(n-1)^2/a(n-2)) - 1.

For n>2, a(n) = 3*a(n-1) - a(n-3).

G.f.: -(x^2-3) / (x^3-3*x+1). - Colin Barker, Dec 13 2012

Mathematica

CoefficientList[Series[- (x^2 - 3)/(x^3 - 3 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 16 2013 *)
RecurrenceTable[{a[1] == 3, a[2] == 9, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 30}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{3, 0, -1}, {3, 9, 26}, 30] (* Harvey P. Dale, Feb 06 2019 *)

Prog

(PARI) T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(3, 9, 30) \\ Colin Barker, Feb 14 2016
(Magma) Tiv:=[3, 9]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..40]]; // Bruno Berselli, Feb 17 2016

Crossrefs

Cf. A076264.

Sequence in context: A303976 A000243 A076264 * A123941 A005774 A273343

Adjacent sequences: A018916 A018917 A018918 * A018920 A018921 A018922

Keyword

nonn,easy

Author

R. K. Guy

Status

approved