A020727 Pisot sequence P(2,7): a(0)=2, a(1)=7, thereafter a(n+1) is the nearest integer to a(n)^2/a(n-1).

2, 7, 24, 82, 280, 956, 3264, 11144, 38048, 129904, 443520, 1514272, 5170048, 17651648, 60266496, 205762688, 702517760, 2398545664, 8189147136, 27959497216, 95459694592, 325919783936, 1112759746560, 3799199418368, 12971278180352, 44286713884672

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

Also Pisot sequence T(2,7). - R. K. Guy

It appears that a(n) = 4*a(n-1) - 2*a(n-2) (holds at least up to n = 1000 but is not known to hold in general).

The recurrence holds up to n = 10^5. - Ralf Stephan, Sep 03 2013

Empirical g.f.: (2-x)/(1-4*x+2*x^2). - Colin Barker, Feb 21 2012

Links

• Colin Barker, Table of n, a(n) for n = 0..1000

Crossrefs

It appears that this is a subsequence of A003480.

See A008776 for definitions of Pisot sequences.

Programs

Magma

Iv:=[2,7]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..30]]; // Bruno Berselli, Feb 04 2016

Mathematica

RecurrenceTable[{a[0] == 2, a[1] == 7, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 04 2016 *)

PARI

pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 2, 7) \\ Colin Barker, Aug 08 2016

Extensions

Edited by N. J. A. Sloane, Aug 17 2009 at the suggestion of R. J. Mathar.