A048584 Pisot sequence L(5,7).

5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Offset

0,1

Comments

a(n)=BA^(n)B(1), n>=0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g. 5=`00`, 7=`010`, 10=`0110`, 15=`01110`,..., in Wythoff code.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Formula

a(n) = Fibonacci(n+4)+2 = A157725(n+4).

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

a(n) = A020743(n-1), n>0. - R. J. Mathar, Oct 15 2008

G.f.: (5 - 3*x - 4*x^2)/((1 - x)*(1 - x - x^2)). - Elmo R. Oliveira, Apr 17 2026

E.g.f.: 2*exp(x) + exp(x/2)*(15*cosh(sqrt(5)*x/2) + 7*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 19 2026

Mathematica

LinearRecurrence[{2, 0, -1}, {5, 7, 10}, 40] (* Harvey P. Dale, Oct 02 2016 *)

Prog

(PARI) pisotL(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]));
  a
}
pisotL(50, 5, 7) \\ Colin Barker, Aug 07 2016

Crossrefs

Subsequence of A018910, A157725.

See A008776 for definitions of Pisot sequences.

Cf. A000201, A001950, A020743, A135817.

Sequence in context: A196936 A188196 A381677 * A085382 A250194 A314293

Adjacent sequences: A048581 A048582 A048583 * A048585 A048586 A048587

Keyword

nonn,easy

Author

David W. Wilson

Status

approved