A020750 Pisot sequence T(5,9), a(n) = floor(a(n-1)^2/a(n-2)).

5, 9, 16, 28, 49, 85, 147, 254, 438, 755, 1301, 2241, 3860, 6648, 11449, 19717, 33955, 58474, 100698, 173411, 298629, 514265, 885608, 1525092, 2626337, 4522773, 7788595, 13412614, 23097646, 39776083, 68497749, 117958865, 203135052, 349815584, 602411753

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

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

Crossrefs

See A008776 for definitions of Pisot sequences.

Programs

Magma

Iv:=[5, 9]; [n le 2 select Iv[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..40]]; // Bruno Berselli, Feb 04 2016

Mathematica

RecurrenceTable[{a[0] == 5, a[1] == 9, a[n] == Floor[a[n - 1]^2/a[n - 2]]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 04 2016 *)
nxt[{a_,b_}]:={b,Floor[b^2/a]}; NestList[nxt,{5,9},40][[All,1]] (* Harvey P. Dale, Aug 29 2021 *)

PARI

pisotT(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));
  a
}
pisotT(50, 5, 9) \\ Colin Barker, Jul 29 2016

Formula

a(n) = 2*a(n-1) - 2*a(n-4) + a(n-5) (holds at least up to n = 40000 but is not known to hold in general).

Note the warning in A010925 from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004: [A010925] and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. - N. J. A. Sloane, Jul 26 2016

Extensions

Deleted unproved recurrence and program based on it. - N. J. A. Sloane, Feb 04 2016