A018920 Pisot sequence T(3,10), a(n) = floor(a(n-1)^2/a(n-2)).

3, 10, 33, 108, 353, 1153, 3766, 12300, 40172, 131202, 428506, 1399501, 4570771, 14928140, 48755311, 159234864, 520061125, 1698519827, 5547366384, 18117700664, 59172417076, 193257136076, 631177877968, 2061427183105, 6732621943159, 21988745758766

Offset: 0

Author

R. K. Guy

Keywords

• nonn

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 Pisot sequences

Crossrefs

See A008776 for definitions of Pisot sequences.

Programs

Magma

Txy:=[3,10]; [n le 2 select Txy[n] else Floor(Self(n-1)^2/Self(n-2)): n in [1..30]]; // Bruno Berselli, Feb 05 2016

Maple

PisotT := proc(a0,a1,n)
    option remember;
    if n = 0 then
        a0 ;
    elif n = 1 then
        a1;
    else
        floor( procname(a0,a1,n-1)^2/procname(a0,a1,n-2)) ;
    end if;
end proc:
A018920 := proc(n)
    PisotT(3,10,n) ;
end proc: # R. J. Mathar, Feb 13 2016

Mathematica

RecurrenceTable[{a[0] == 3, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] ]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)

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, 3, 10) \\ Colin Barker, Jul 29 2016

Formula

a(n) = 3*a(n-1) + a(n-2) - a(n-4) - a(n-5) - a(n-6) (holds at least up to n = 1000 but is not known to hold in general).

Extensions

Corrected by David W. Wilson