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

Author

R. K. Guy

Keywords

• nonn
• easy

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

Crossrefs

Cf. A008937.

Programs

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

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 *)

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

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

Extensions

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