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%I #65 Jan 25 2023 11:47:26
%S 2,6,8,14,22,36,58,94,152,246,398,644,1042,1686,2728,4414,7142,11556,
%T 18698,30254,48952,79206,128158,207364,335522,542886,878408,1421294,
%U 2299702,3720996,6020698,9741694,15762392,25504086,41266478,66770564,108037042
%N Fibonacci sequence beginning 2, 6.
%C For n>=3 the number of perfect matchings in the n-antiprism graph. - _Andrew Howroyd_, May 17 2017
%H Reinhard Zumkeller, <a href="/A022112/b022112.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Yun-Tak Oh, Hosho Katsura, Hyun-Yong Lee and Jung Hoon Han, <a href="https://arxiv.org/abs/1709.01344">Proposal of a spin-one chain model with competing dimer and trimer interactions</a>, arXiv:1709.01344 [cond-mat.str-el], 2017.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AntiprismGraph.html">Antiprism Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependentEdgeSet.html">Independent Edge Set</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Matching.html">Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectMatching.html">Perfect Matching</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MaximumIndependentEdgeSet.html">Maximum Independent Edge Set</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F a(n) = 4*Fibonacci(n+2) - 2*Fibonacci(n+1). - _Gary Detlefs_, Dec 21 2010
%F a(n) = 2*A000204(n+1). - _R. J. Mathar_, Mar 11 2011
%F G.f.: ( -2-4*x ) / ( -1+x+x^2 ). - _R. J. Mathar_, Mar 11 2011
%F a(n) = Fibonacci(n-2) + Fibonacci(n+4). - _Gary Detlefs_, Mar 31 2012
%F a(n) = L(n - 1) + L(n) + L(n + 1), for n > 0, where L(n) is the n-th Lucas number (A000204). - _Alonso del Arte_, Sep 25 2013
%F a(n) = L(n + 3) - L(n). - _Bruno Berselli_, Jun 15 2017
%F From _Colin Barker_, Oct 27 2017: (Start)
%F a(n) = (2^(-n)*((1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))) / sqrt(5).
%F a(n) = a(n-1) + a(n-2) for > 1.
%F (End)
%t LinearRecurrence[{1, 1}, {2, 6}, 40] (* _Harvey P. Dale_, Apr 21 2012 *)
%t 2 LucasL[Range[30]] (* _Alonso del Arte_, Sep 25 2013 *)
%o (Haskell)
%o a022112 n = a022112_list !! n
%o a022112_list = 2 : 6 : zipWith (+) (tail a022112_list) a022112_list
%o -- _Reinhard Zumkeller_, Apr 08 2012
%o (PARI) a(n)=4*fibonacci(n+2)-2*fibonacci(n+1) \\ _Charles R Greathouse IV_, Oct 07 2015
%o (PARI) Vec(2*(1 + 2*x) / (1 - x - x^2) + O(x^60)) \\ _Colin Barker_, Oct 27 2017
%Y Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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