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 A087131 a(n) = 2^n*Lucas(n), where Lucas = A000032. 16
 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) 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 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 237. LINKS L. E. 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 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 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]] (* 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)  cat [2^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017 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 August 6 09:38 EDT 2020. Contains 336245 sequences. (Running on oeis4.)