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%I #50 Jan 08 2020 09:27:26
%S 2,3,10,29,82,227,618,1661,4418,11651,30506,79389,205522,529635,
%T 1359434,3476989,8865026,22538755,57157578,144615709,365127634,
%U 920110051,2314564522,5812911741,14576950082,36503608707,91294323178,228049363229,569017421650,1418290058723
%N Number of matchings in the n-helm graph.
%C Extended to a(0)-a(2) using the formula.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm 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 (4,-2,-4,-1).
%F a(n) = ((1-sqrt(2))^n*(4-sqrt(2)*n)+(1+sqrt(2))^n*(4+sqrt(2)*n))/4;
%F a(n) = A002203(n) + n*A000129(n).
%F a(n) = 4*a(n-1)-2*a(n-2)-4*a(n-3)-a(n-4).
%F G.f.: (2-5*x+2*x^2+3*x^3)/(-1+2*x+x^2)^2.
%t Table[1/4 ((1 - Sqrt[2])^n (4 - Sqrt[2] n) + (1 + Sqrt[2])^n (4 + Sqrt[2] n)), {n, 0, 20}] // Expand
%t Table[LucasL[n, 2] + n Fibonacci[n, 2], {n, 0, 20}]
%t LinearRecurrence[{4, -2, -4, -1}, {3, 10, 29, 82}, {0, 20}]
%t CoefficientList[Series[(2 - 5 x + 2 x^2 + 3 x^3)/(-1 + 2 x + x^2)^2, {x, 0, 20}], x]
%K nonn,easy
%O 0,1
%A _Eric W. Weisstein_, May 27 2017