A048575 Pisot sequences L(2,5), E(2,5).

2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

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
• Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M. On the cyclically fully commutative elements of Coxeter groups, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Section 5.1
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-1)

Crossrefs

Subsequence of A001519. See A008776 for definitions of Pisot sequences.

Programs

Magma

[Fibonacci(2*n+3): n in [0..40]]; // Vincenzo Librandi, Jul 12 2015

Mathematica

LinearRecurrence[{3, -1}, {2, 5}, 40] (* Vincenzo Librandi, Jul 12 2015 *)

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, 2, 5) \\ Colin Barker, Jul 27 2016

Formula

a(n) = A000045(2n+3). a(n) = 3a(n-1) - a(n-2).

G.f.: (2-x)/(1-3x+x^2). [Philippe Deléham, Nov 16 2008]

a(n) = 2*A001906(n+1)-A001906(n). - R. J. Mathar, Jun 11 2019