A010919 Pisot sequence T(4,13), a(n) = floor(a(n-1)^2/a(n-2)).

4, 13, 42, 135, 433, 1388, 4449, 14260, 45706, 146496, 469546, 1504979, 4823727, 15460908, 49554976, 158832563, 509086778, 1631714194, 5229935889, 16762880107, 53728029453, 172207945799, 551957272549, 1769121798104, 5670351840955, 18174492018967

Offset: 0

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• D. W. Boyd, Pisot sequences which satisfy no linear recurrences, Acta Arith. 32 (1) (1977) 89-98
• D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305
• D. W. Boyd, On linear recurrence relations satisfied by Pisot sequences, Acta Arithm. 47 (1) (1986) 13
• D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191
• D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993; with updates from 1996 and 1999.
• D. G. Cantor, On families of Pisot E-sequences, Ann. Sci. Ecole Nat. Sup. 9 (2) (1976) 283-308

Crossrefs

Cf. A022029, A010925.

Programs

Mathematica

a[0] = 4; a[1] = 13; a[n_] := a[n] = Floor[a[n-1]^2/a[n-2]]; Array[a, 30, 0] (* Jean-François Alcover, Dec 14 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, 4, 13) \\ Colin Barker, Jul 29 2016

Formula

Appears to satisfy the g.f. (4+x-x^2-x^4-x^36)/(1-3*x-x^2+x^3+x^5+x^37), where there is a common factor of 1+x that can be canceled, so the sequence appears to satisfy a linear recurrence of order 36. I believe that David Boyd has proved that the sequence does indeed satisfy this recurrence. - N. J. A. Sloane, Aug 11 2016