|
%I #18 Sep 08 2022 08:46:19
%S 1,2,5,11,24,51,109,234,503,1081,2322,4987,10711,23006,49415,106139,
%T 227976,489669,1051759,2259072,4852259,10422163,22385754,48082339,
%U 103276009,221826440,476460797,1023389687,2198137722,4721377893,10141043023,21781936530
%N Number of maximal matchings in the ladder graph P_2 X P_n.
%H Andrew Howroyd, <a href="/A286945/b286945.txt">Table of n, a(n) for n = 1..500</a>
%H Svenja Huntemann, Neil A. McKay, <a href="https://arxiv.org/abs/1909.12419">Counting Domineering Positions</a>, arXiv:1909.12419 [math.CO], 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder 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/MaximalIndependentEdgeSet.html">Maximal Independent Edge Set</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,0,1,1).
%F a(n) = 2*a(n-1) + a(n-4) + a(n-5) for n>5.
%F G.f.: x*(1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)).
%p seq(coeff(series(x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5), x, n+1), x, n), n = 1..35); # _G. C. Greubel_, Dec 30 2019
%t Table[3Cos[nPi/3]/13 - 5Sin[nPi/3]/(13 Sqrt[3]) + RootSum[-1 -2# -#^2 +#^3 &, (-6 -72# +80#^2) #^n &]/403, {n, 35}] (* _Eric W. Weisstein_, Jul 13 2017 *)
%t LinearRecurrence[{2,0,0,1,1}, {1,2,5,11,24}, 35] (* _Eric W. Weisstein_, Jul 13 2017 *)
%t CoefficientList[Series[(1+x^2+x^3+x^4)/(1-2x-x^4-x^5), {x, 0, 35}], x] (* _Eric W. Weisstein_, Jul 13 2017 *)
%o (PARI) Vec((1+x^2+x^3+x^4)/((1-x+x^2)*(1-x-2*x^2-x^3)) + O(x^35))
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 35); Coefficients(R!( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) )); // _G. C. Greubel_, Dec 30 2019
%o (Sage)
%o def A286945_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x*(1+x^2+x^3+x^4)/(1-2*x-x^4-x^5) ).list()
%o a=A286945_list(35); a[1:] # _G. C. Greubel_, Dec 30 2019
%o (GAP) a:=[1,2,5,11,24];; for n in [6..35] do a[n]:=2*a[n-1]+a[n-4]+a[n-5]; od; a; # _G. C. Greubel_, Dec 30 2019
%Y Cf. A284703, A284710, A286911.
%K nonn
%O 1,2
%A _Andrew Howroyd_, May 16 2017
|
