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A087131 a(n) = 2^n*Lucas(n), where Lucas = A000032. 15
2, 2, 12, 32, 112, 352, 1152, 3712, 12032, 38912, 125952, 407552, 1318912, 4268032, 13811712, 44695552, 144637952, 468058112, 1514668032, 4901568512, 15861809152, 51329892352, 166107021312, 537533612032, 1739495309312 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Number of ways to tile an n-bracelet with two types of colored squares and four types of colored dominoes.

Inverse binomial transform of even Lucas numbers (A014448).

From L. Edson Jeffery, Apr 25 2011 (Start):

Let A be the unit-primitive matrix (see [Jeffery])

A=A_(10,4)=

(0 0 0 0 1)

(0 0 0 2 0)

(0 0 2 0 1)

(0 2 0 2 0)

(2 0 2 0 1).

Then a(n)=(Trace(A^n)-1)/2. Also a(n)=Trace((2*A_(5,1))^n), where A_(5,1)=[(0,1); (1,1)] is also a unit-primitive matrix. (End)

For n >= 3, also the number of connected dominating sets in the n-sun graph. - Eric W. Weisstein, May 02 2017

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 237.

LINKS

Table of n, a(n) for n=0..24.

L. E. Jeffery, Unit-primitive matrices

Eric Weisstein's World of Mathematics, Connected Dominating Set

Eric Weisstein's World of Mathematics, Sun Graph

Index entries for linear recurrences with constant coefficients, signature (2,4).

FORMULA

Recurrence: a(n) = 2a(n-1) + 4a(n-2), a(0)=2, a(1)=2.

G.f.: 2(1-x) / (1-2x-4x^2).

a(n) = (1+sqrt(5))^n + (1-sqrt(5))^n.

For n>=2, a(n) = Trace of matrix [({2,2},{2,0})^n]. - Artur Jasinski, Jan 09 2007

a(n) = 2*[A063727(n)-A063727(n-1)]. - R. J. Mathar, Nov 16 2007

a(n) = (5*A052899(n)-1)/2. - L. Edson Jeffery, Apr 25 2011

a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 5*x^2) )^n for n >= 1. - Peter Bala, Jun 23 2015

MATHEMATICA

Table[Tr[MatrixPower[{{2, 2}, {2, 0}}, x]], {x, 1, 20}] (* Artur Jasinski, Jan 09 2007 *)

Table[2^n LucasL[n], {n, 20}] (* Eric W. Weisstein, May 02 2017 *)

2^# LucasL[#] & [Range[20]] (* Eric W. Weisstein, May 02 2017 *)

PROG

(Sage) [lucas_number2(n, 2, -4) for n in xrange(0, 25)] # Zerinvary Lajos, Apr 30 2009

CROSSREFS

Equals 2*A084057(n). First differences of A006483 and A103435.

Sequence in context: A185144 A185344 A237275 * A199240 A173842 A131444

Adjacent sequences:  A087128 A087129 A087130 * A087132 A087133 A087134

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Aug 16 2003

EXTENSIONS

Edited by Ralf Stephan, Feb 08 2005

STATUS

approved

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