A010924 Pisot sequence E(8,55), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

8, 55, 378, 2598, 17856, 122724, 843480, 5797224, 39844224, 273848688, 1882157472, 12936036960, 88909166592, 611071221312, 4199882327424, 28865721292416, 198393621719040, 1363556058068736, 9371698078726656, 64411524820772352, 442699337396994048

Offset

0,1

Links

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

D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.

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.

Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

S. B. Ekhad, N. J. A. Sloane, and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (6, 6).

Formula

Conjecture: a(n) = 6*a(n-1) + 6*a(n-2), n > 1; a(0)=8, a(1)=55; g.f.: (8+7x)/(1-6x-6x^2). - Philippe Deléham, Nov 19 2008

Theorem: a(n) = 6*a(n-1) + 6*a(n-2) for n >= 2. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016

Mathematica

a[0] = 8; a[1] = 55; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 20}] (* Michael De Vlieger, Jul 27 2016 *)
LinearRecurrence[{6, 6}, {8, 55}, 30] (* Harvey P. Dale, Mar 06 2022 *)

Prog

(PARI) pisotE(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]+1/2));
  a
}
pisotE(50, 8, 55) \\ Colin Barker, Jul 27 2016

Crossrefs

Sequence in context: A143420 A075734 A033890 * A308687 A010918 A019484

Adjacent sequences: A010921 A010922 A010923 * A010925 A010926 A010927

Keyword

nonn,easy

Author

Simon Plouffe

Status

approved