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

0,1

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

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

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 *)

Prog

(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

Crossrefs

Cf. A022029, A010925.

Sequence in context: A192910 A289807 A022029 * A277667 A274952 A010900

Adjacent sequences: A010916 A010917 A010918 * A010920 A010921 A010922

Keyword

nonn

Author

Simon Plouffe and N. J. A. Sloane.

Status

approved