A020706 Pisot sequences L(4,6), E(4,6).

4, 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

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Subsequence of A001611, A048577. See A008776 for definitions of Pisot sequences.

Programs

Magma

I:=[4, 6, 9]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Apr 20 2012

Mathematica

CoefficientList[Series[(4-2*x-3*x^2)/(1-x)/(1-x-x^2),{x,0,40}],x](* Vincenzo Librandi, Apr 20 2012 *)

Formula

a(n) = Fib(n+4)+1 = A000045(n+4)+1.

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

G.f.: (4-2*x-3*x^2)/(1-x)/(1-x-x^2). - Colin Barker, Feb 21 2012