A083858 Expansion of x/(1 - 3*x - 6*x^2).

0, 1, 3, 15, 63, 279, 1215, 5319, 23247, 101655, 444447, 1943271, 8496495, 37149111, 162426303, 710173575, 3105078543, 13576277079, 59359302495, 259535569959, 1134762524847, 4961500994295, 21693078131967, 94848240361671

Offset

0,3

Comments

Binomial transform of A015443. A row of array A083857.

Pisano period lengths: 1, 1, 1, 1, 12, 1, 8, 1, 1, 12, 110, 1, 168, 8, 12, 2, 16, 1, 360, 12, ... - R. J. Mathar, Aug 10 2012

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,6).

Formula

a(n) = 3*a(n-1) + 6*a(n-2), a(0)=0, a(1)=1.

a(n) = (3*sqrt(33)/2 + 21/2)^(n/2)/sqrt(33) - (21/2 - 3*sqrt(33)/2)^(n/2)*(-1)^n/sqrt(33).

G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(6*k+3 + 6*x )/( x*(6*k+6 + 6*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013

a(n) = B(n, k + 2^(n-1)) - B(n,k) where B(n,k) is formed by the family of recursions b(n) = 3*(b(n-1) + b(n-2))/2, with b(0) = 1 and b(1) = k, as explained further in A249861. - Richard R. Forberg, Nov 04 2014

Mathematica

a[n_]:=(MatrixPower[{{1, 2}, {1, -4}}, n].{{1}, {1}})[[2, 1]]; Table[Abs[a[n]], {n, -1, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3, 6}, {0, 1}, 30] (* G. C. Greubel, Jan 16 2018 *)

Prog

(Sage) [lucas_number1(n, 3, -6) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009
(PARI) x='x+O('x^30); concat([0], Vec(x/(1-3*x-6*x^2))) \\ G. C. Greubel, Jan 16 2018
(Magma) I:=[0, 1]; [n le 2 select I[n] else 3*Self(n-1) + 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Crossrefs

Cf. A015523, A015524.

Sequence in context: A024036 A111303 A118339 * A151241 A080948 A098102

Adjacent sequences: A083855 A083856 A083857 * A083859 A083860 A083861

Keyword

easy,nonn

Author

Paul Barry, May 06 2003

Status

approved