A010904 Pisot sequence E(4,14): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=14.

4, 14, 49, 172, 604, 2121, 7448, 26154, 91841, 322504, 1132488, 3976785, 13964668, 49037590, 172197809, 604680724, 2123364868, 7456295833, 26183134320, 91943310482, 322863269121, 1133749589840, 3981215131600, 13980224615841, 49092217790004, 172389637059934

Offset

0,1

References

Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

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.

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

Formula

Theorem: a(0)=4, a(1)=14, a(2)=49; for n>2, a(n) = 4*a(n-1)-2*a(n-2)+a(n-3). Proved using the PtoRv program of Ekhad-Sloane-Zeilberger. (Conjectured by Harvey P. Dale, May 02 2012.) - N. J. A. Sloane, Sep 09 2016

Mathematica

RecurrenceTable[{a[0] == 4, a[1] == 14, a[n] == Floor[a[n-1]^2/a[n-2] + 1/2]}, a , {n, 30}] (* Harvey P. Dale, May 02 2012 *)

Prog

(Magma) I:=[4, 14]; [n le 2 select I[n] else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..25]]; // Bruno Berselli, Sep 03 2013
(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, 4, 14) \\ Colin Barker, Jul 27 2016

Crossrefs

Sequence in context: A278692 A071733 A291384 * A277084 A071737 A071741

Adjacent sequences: A010901 A010902 A010903 * A010905 A010906 A010907

Keyword

nonn

Author

Simon Plouffe

Status

approved