A018921 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(4,8).

4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840, 216847936, 398845537, 733591314, 1349284788, 2481721640

Offset

0,1

Comments

Not to be confused with the Pisot T(4,8) sequence, which is A020707. - R. J. Mathar, Feb 13 2016

Links

Colin Barker, 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 (2,0,0,-1).

Index entries for Pisot sequences

Formula

a(n) = 2*a(n-1) - a(n-4).

G.f.: (4-x^2-2*x^3) / ((1-x)*(1-x-x^2-x^3)). - Colin Barker, Feb 08 2012

a(n) = A008937(n+3) = A027084(n+3)+1. [first index correct by R. J. Mathar, Jun 24 2020]

a(n) = 2*a(n-1) - A008937(n). - Vincenzo Librandi, Feb 12 2016

Mathematica

RecurrenceTable[{a[1] == 4, a[2] == 8, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{2, 0, 0, -1}, {4, 8, 15, 28}, 40] (* Harvey P. Dale, Mar 05 2019 *)

Prog

(PARI) Vec((4-x^2-2*x^3)/((1-x)*(1-x-x^2-x^3)) + O(x^40)) \\ Colin Barker, Feb 13 2016
(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(4, 8, 30) \\ Colin Barker, Feb 14 2016
(Magma) Tiv:=[4, 8]; [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. A008937.

Sequence in context: A348897 A349201 A301203 * A305218 A321500 A103536

Adjacent sequences: A018918 A018919 A018920 * A018922 A018923 A018924

Keyword

nonn,easy

Author

R. K. Guy

Extensions

Comments moved to formula, and typo in data fixed by Colin Barker, Feb 13 2016

Status

approved