A048575 - OEIS

%I #39 Dec 30 2023 23:47:39

%S 2,5,13,34,89,233,610,1597,4181,10946,28657,75025,196418,514229,

%T 1346269,3524578,9227465,24157817,63245986,165580141,433494437,

%U 1134903170,2971215073,7778742049,20365011074,53316291173,139583862445,365435296162,956722026041

%N Pisot sequences L(2,5), E(2,5).

%D Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.

%H Colin Barker, <a href="/A048575/b048575.txt">Table of n, a(n) for n = 0..1000</a>

%H Boothby, T.; Burkert, J.; Eichwald, M.; Ernst, D. C.; Green, R. M.; Macauley, M.  <a href="https://doi.org/10.1007/s10801-011-0327-z">On the cyclically fully commutative elements of Coxeter groups</a>, J. Algebr. Comb. 36, No. 1, 123-148 (2012), Section 5.1

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1)

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

%F G.f.: (2-x)/(1-3x+x^2). [_Philippe Deléham_, Nov 16 2008]

%F a(n) = 2*A001906(n+1)-A001906(n). - _R. J. Mathar_, Jun 11 2019

%t LinearRecurrence[{3, -1}, {2, 5}, 40] (* _Vincenzo Librandi_, Jul 12 2015 *)

%o (Magma) [Fibonacci(2*n+3): n in [0..40]]; // _Vincenzo Librandi_, Jul 12 2015

%o (PARI) pisotE(nmax, a1, a2) = {

%o   a=vector(nmax); a[1]=a1; a[2]=a2;

%o   for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));

%o   a

%o }

%o pisotE(50, 2, 5) \\ _Colin Barker_, Jul 27 2016

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

%K nonn,easy

%O 0,1

%A _David W. Wilson_