A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15.

4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244

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.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (4,-1).

Formula

a(n) = 4*a(n-1) - a(n-2) for n>=2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016

This was conjectured by Colin Barker, Apr 16 2012, and implies the G.f.: (4-x)/(1-4*x+x^2) and the formula a(n) = ((1+sqrt(3))^(2*n+4)-(1-sqrt(3))^(2*n+4))/(2^(n+3)*sqrt(3)).

Partial sums of A079935. - Erin Pearse, Dec 13 2018

Mathematica

a[0] = 4; a[1] = 15; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 24}] (* Michael De Vlieger, Jul 27 2016 *)

Prog

(Magma) /* By definition: */ [n le 2 select 11*n-7 else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..22]]; // Bruno Berselli, Apr 16 2012
(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, 15) \\ Colin Barker, Jul 27 2016
(Sage)
@cached_function
def A010905(n):
    if n==0: return 4
    elif n==1: return 15
    else: return 4*A010905(n-1) - A010905(n-2)
[A010905(n) for n in range(30)] # G. C. Greubel, Dec 13 2018

Crossrefs

Cf. A010925, A001353, A079935, A195503.

Sequence in context: A125905 A195503 A001353 * A026030 A047038 A158500

Adjacent sequences: A010902 A010903 A010904 * A010906 A010907 A010908

Keyword

nonn

Author

Simon Plouffe

Extensions

Edited by N. J. A. Sloane, Jul 26 2016 and Sep 09 2016

Status

approved