The On-Line Encyclopedia of Integer Sequences (OEIS)

A048575 revision #36

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

(list; graph; refs; listen; history; text; internal format)

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

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)

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) = [(3/2)+(1/2)*sqrt(5)]^n+(2/5)*[(3/2)+(1/2)*sqrt(5)]^n*sqrt(5)-(2/5)*[(3/2)-(1/2)*sqrt(5)]^n *sqrt(5)+[(3/2)-(1/2)*sqrt(5)]^n, with n>=0. [Paolo P. Lava, Nov 20 2008]

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

MATHEMATICA

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

PROG

(MAGMA) [Fibonacci(2*n+3): n in [0..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));

}

pisotE(50, 2, 5) \\ Colin Barker, Jul 27 2016

CROSSREFS

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