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%I #44 Feb 14 2024 10:15:10
%S 2,5,17,56,185,611,2018,6665,22013,72704,240125,793079,2619362,
%T 8651165,28572857,94369736,311682065,1029415931,3399929858,
%U 11229205505,37087546373,122491844624,404563080245,1336181085359,4413106336322,14575500094325,48139606619297
%N a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
%C Inverse binomial transform of A052961 (without the leading 1).
%C For n >= 1, also the number of matchings in the n-alkane graph. - _Eric W. Weisstein_, Jul 14 2021
%H Alois P. Heinz, <a href="/A180148/b180148.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlkaneGraph.html">Alkane Graph</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/MaximumIndependentEdgeSet.html">Maximum Independent Edge Set</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,1).
%F G.f.: (2-x)/(1-3*x-x^2).
%F a(n) = 3*a(n-1) + a(n-2) with a(0)=2 and a(1)=5.
%F a(n) = ((4+7*A)*A^(-n-1) + (4+7*B)*B^(-n-1))/13 with A = (-3+sqrt(13))/2 and B = (-3-sqrt(13))/2.
%F Lim_{k->infinity} a(n+k)/a(k) = (-1)^n*2/(A006497(n) - A006190(n)*sqrt(13)).
%F a(n) = 2 * Sum_{k=0..n-2} A168561(n-2,k)*3^k + 5 * Sum_{k=0..n-1} A168561(n-1,k)*3^k, n>0. - _R. J. Mathar_, Feb 14 2024
%F a(n) = 2*A006190(n+1) - A006190(n). - _R. J. Mathar_, Feb 14 2024
%p a:= n-> (<<0|1>, <1|3>>^n. <<2, 5>>)[1,1]:
%p seq(a(n), n=0..27); # _Alois P. Heinz_, Jul 14 2021
%t LinearRecurrence[{3, 1}, {5, 7}, 20] (* _Eric W. Weisstein_, Jul 14 2021 *)
%t CoefficientList[Series[(2 - x)/(1 - 3 x - x^2), {x, 0, 20}], x] (* _Eric W. Weisstein_, Jul 14 2021 *)
%o (PARI) a(n)=([0,1;1,3]^n*[2;5])[1,1] \\ _Charles R Greathouse IV_, Oct 13 2016
%Y Cf. A003688, A052906.
%Y Appears in A180142.
%Y Cf. A000602 (more information on n-alkanes).
%K easy,nonn
%O 0,1
%A _Johannes W. Meijer_, Aug 13 2010
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