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A083858 Expansion of x/(1-3x-6x^2). 15
0, 1, 3, 15, 63, 279, 1215, 5319, 23247, 101655, 444447, 1943271, 8496495, 37149111, 162426303, 710173575, 3105078543, 13576277079, 59359302495, 259535569959, 1134762524847, 4961500994295, 21693078131967, 94848240361671 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..23.

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

FORMULA

a(n)=3a(n-1)+6a(n-2), a(0)=0, a(1)=1; a(n)=(3sqrt(33)/2+21/2)^(n/2)/sqrt(33)-(21/2-3sqrt(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 *)

PROG

(Sage) [lucas_number1(n, 3, -6) for n in xrange(0, 24)] # Zerinvary Lajos, Apr 22 2009

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

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Last modified October 20 17:39 EDT 2017. Contains 293648 sequences.