A087131 a(n) = 2^n*Lucas(n), where Lucas = A000032.

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

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)

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

Also the number of total dominating sets in the n-sun graph for n >= 3. - Eric W. Weisstein, Apr 27 2018

References

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, identity 237, p. 132.

Links

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

L. Edson Jeffery, Unit-primitive matrices

Eric Weisstein's World of Mathematics, Connected Dominating Set.

Eric Weisstein's World of Mathematics, Sun Graph.

Eric Weisstein's World of Mathematics, Total Dominating Set.

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

Formula

a(n) = 2*A084057(n).

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

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

Sum_{n>=1} 1/a(n) = (1/2) * A269992. - Amiram Eldar, Nov 17 2020

From Amiram Eldar, Jan 15 2022: (Start)

a(n) == 2 (mod 10).

a(n) = 5 * A014334(n) + 2.

a(n) = 10 * A014335(n) + 2. (End)

Mathematica

Table[Tr[MatrixPower[{{2, 2}, {2, 0}}, x]], {x, 1, 20}] (* Artur Jasinski, Jan 09 2007 *)
Join[{2}, Table[2^n LucasL[n], {n, 20}]] (* Eric W. Weisstein, May 02 2017 *)
Join[{2}, 2^# LucasL[#] & [Range[20]]] (* Eric W. Weisstein, May 02 2017 *)
LinearRecurrence[{2, 4}, {2, 12}, {0, 20}] (* Eric W. Weisstein, Apr 27 2018 *)
CoefficientList[Series[(2 (-1 + x))/(-1 + 2 x + 4 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 27 2018 *)

Prog

(Sage) [lucas_number2(n, 2, -4) for n in range(0, 25)] # Zerinvary Lajos, Apr 30 2009
(PARI) for(n=0, 30, print1(if(n==0, 2, 2^n*(fibonacci(n+1) + fibonacci(n-1))), ", ")) \\ G. C. Greubel, Dec 18 2017
(PARI) first(n) = Vec(2*(1-x)/(1-2*x-4*x^2) + O(x^n)) \\ Iain Fox, Dec 19 2017
(Magma) [2] cat [2^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017

Crossrefs

First differences of A006483 and A103435.

Cf. A000032, A014334, A014335, A084057, A269992.

Sequence in context: A185144 A185344 A237275 * A199240 A349529 A173842

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