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%I #64 Sep 08 2022 08:45:11
%S 2,2,12,32,112,352,1152,3712,12032,38912,125952,407552,1318912,
%T 4268032,13811712,44695552,144637952,468058112,1514668032,4901568512,
%U 15861809152,51329892352,166107021312,537533612032,1739495309312
%N a(n) = 2^n*Lucas(n), where Lucas = A000032.
%C Number of ways to tile an n-bracelet with two types of colored squares and four types of colored dominoes.
%C Inverse binomial transform of even Lucas numbers (A014448).
%C From _L. Edson Jeffery_, Apr 25 2011: (Start)
%C Let A be the unit-primitive matrix (see [Jeffery])
%C A=A_(10,4)=
%C (0 0 0 0 1)
%C (0 0 0 2 0)
%C (0 0 2 0 1)
%C (0 2 0 2 0)
%C (2 0 2 0 1).
%C 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)
%C Also the number of connected dominating sets in the n-sun graph for n >= 3. - _Eric W. Weisstein_, May 02 2017
%C Also the number of total dominating sets in the n-sun graph for n >= 3. - _Eric W. Weisstein_, Apr 27 2018
%D Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, identity 237, p. 132.
%H L. Edson Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConnectedDominatingSet.html">Connected Dominating Set</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SunGraph.html">Sun Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotalDominatingSet.html">Total Dominating Set</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,4).
%F a(n) = 2*A084057(n).
%F Recurrence: a(n) = 2a(n-1) + 4a(n-2), a(0)=2, a(1)=2.
%F G.f.: 2*(1-x)/(1-2*x-4*x^2).
%F a(n) = (1+sqrt(5))^n + (1-sqrt(5))^n.
%F For n>=2, a(n) = Trace of matrix [({2,2},{2,0})^n]. - _Artur Jasinski_, Jan 09 2007
%F a(n) = 2*[A063727(n)-A063727(n-1)]. - _R. J. Mathar_, Nov 16 2007
%F a(n) = (5*A052899(n)-1)/2. - _L. Edson Jeffery_, Apr 25 2011
%F a(n) = [x^n] ( 1 + x + sqrt(1 + 2*x + 5*x^2) )^n for n >= 1. - _Peter Bala_, Jun 23 2015
%F Sum_{n>=1} 1/a(n) = (1/2) * A269992. - _Amiram Eldar_, Nov 17 2020
%F From _Amiram Eldar_, Jan 15 2022: (Start)
%F a(n) == 2 (mod 10).
%F a(n) = 5 * A014334(n) + 2.
%F a(n) = 10 * A014335(n) + 2. (End)
%t Table[Tr[MatrixPower[{{2, 2}, {2, 0}}, x]], {x, 1, 20}] (* _Artur Jasinski_, Jan 09 2007 *)
%t Join[{2}, Table[2^n LucasL[n], {n, 20}]] (* _Eric W. Weisstein_, May 02 2017 *)
%t Join[{2}, 2^# LucasL[#] & [Range[20]]] (* _Eric W. Weisstein_, May 02 2017 *)
%t LinearRecurrence[{2, 4}, {2, 12}, {0, 20}] (* _Eric W. Weisstein_, Apr 27 2018 *)
%t CoefficientList[Series[(2 (-1 + x))/(-1 + 2 x + 4 x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Apr 27 2018 *)
%o (Sage) [lucas_number2(n,2,-4) for n in range(0, 25)] # _Zerinvary Lajos_, Apr 30 2009
%o (PARI) for(n=0,30, print1(if(n==0, 2, 2^n*(fibonacci(n+1) + fibonacci(n-1))), ", ")) \\ _G. C. Greubel_, Dec 18 2017
%o (PARI) first(n) = Vec(2*(1-x)/(1-2*x-4*x^2) + O(x^n)) \\ _Iain Fox_, Dec 19 2017
%o (Magma) [2] cat [2^n*Lucas(n): n in [1..30]]; // _G. C. Greubel_, Dec 18 2017
%Y First differences of A006483 and A103435.
%Y Cf. A000032, A014334, A014335, A084057, A269992.
%K easy,nonn
%O 0,1
%A _Paul Barry_, Aug 16 2003
%E Edited by _Ralf Stephan_, Feb 08 2005
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