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%I #22 Oct 02 2017 19:00:37
%S 4,13,42,131,398,1186,3482,10103,29034,82777,234424,660098,1849552,
%T 5160001,14341098,39723791,109701122,302131618,830079014,2275509227,
%U 6225274794,16999389733,46341292012,126130604546,342800478748,930414584821,2522124577962,6828859302683
%N Number of matchings in the n-gear graph.
%H Colin Barker, <a href="/A287349/b287349.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GearGraph.html">Gear 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F a(n) = Lucas(2*n) + n*Fibonacci(2*n) for n > 0.
%F G.f.: x*(4 - 11*x + 8*x^2 - 2*x^3)/(1 - 3*x + x^2)^2. - _Ilya Gutkovskiy_, May 23 2017
%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4) for n>4. - _Colin Barker_, Jun 05 2017
%t Table[LucasL[2 n] + n Fibonacci[2 n], {n, 20}]
%t LinearRecurrence[{6, -11, 6, -1}, {4, 13, 42, 131}, 30]
%t CoefficientList[Series[(42 - 121 x + 74 x^2 - 13 x^3)/(1 - 3 x + x^2)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Oct 02 2017 *)
%o (Python)
%o from sympy import lucas, fibonacci
%o def a(n): return lucas(2*n) + n*fibonacci(2*n) # _Indranil Ghosh_, May 24 2017
%o (PARI) Vec(x*(4 - 11*x + 8*x^2 - 2*x^3)/(1 - 3*x + x^2)^2 + O(x^30)) \\ _Colin Barker_, Jun 05 2017
%o (PARI) a(n) = fibonacci(2*n-1) + n*fibonacci(2*n) + fibonacci(2*n+1); \\ _Altug Alkan_, Oct 02 2017
%Y Cf. A000032, A000045.
%K nonn,easy
%O 1,1
%A _Eric W. Weisstein_, May 23 2017