A020717 Pisot sequences L(6,9), E(6,9).

6, 9, 14, 22, 35, 56, 90, 145, 234, 378, 611, 988, 1598, 2585, 4182, 6766, 10947, 17712, 28658, 46369, 75026, 121394, 196419, 317812, 514230, 832041, 1346270, 2178310, 3524579, 5702888, 9227466, 14930353, 24157818, 39088170, 63245987, 102334156, 165580142

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
• Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
• Index entries for Pisot sequences
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Subsequence of A001611, A048577.

See A008776 for definitions of Pisot sequences.

Pairwise sums of A018910.

Programs

Mathematica

Table[Fibonacci[n + 5] + 1, {n, 0, 36}] (* Michael De Vlieger, Jul 27 2016 *)

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, 6, 9) \\ Colin Barker, Jul 27 2016

Formula

a(n) = Fibonacci(n+5)+1 = A001611(n+5).

a(n) = 2*a(n-1) - a(n-3).

a(n) = A020706(n+1). - R. J. Mathar, Oct 25 2008