A102436 Number of matchings of the corona L'(n) of the ladder graph L(n)=P_2 X P_n. and the complete graph K(1); in other words, L'(n) is the graph constructed from L(n) by adding for each vertex v a new vertex v' and the edge vv'.
1, 5, 34, 223, 1469, 9672, 63685, 419329, 2761042, 18179883, 119704137, 788183312, 5189736537, 34171448333, 224999452834, 1481492773799, 9754783005797, 64229669677144, 422915657312253, 2784657839576297, 18335379997029650
Offset: 0
Examples
a(2)=34 because in the graph L'(2) with vertex set {A,B,C,D,a,b,c,d} and edge set {AB,BC,CD,AD,Aa,Bb,Cc,Dd} we have one 0-matching (the empty set), eight 1-matchings (each edge as a singleton), sixteen 2-matchings (see Example in A102435), eight 3-matchings (any 3-element subset of {Aa,Bb,Cc,Dd} and {Aa,Bb,CD},{Bb,Cc,AD},{Cc,Dd,AB},{Aa,Dd,BC}) and one 4-matching ({Aa,Bb,Cc,Dd}).
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- M. Katz, C. Stenson, Tiling a 2xn-board with squares and dominoes, J. Int. Seq. 12 (2009) # 09.2.2
- Index entries for linear recurrences with constant coefficients, signature (6,4,-1).
Programs
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GAP
a:=[1,5,34];; for n in [4..30] do a[n]:=6*a[n-1]+4*a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 27 2019
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Magma
I:=[1,5,34]; [n le 3 select I[n] else 6*Self(n-1) +4*Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 27 2019
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Maple
a[0]:=1: a[1]:=5: a[2]:=34: for n from 3 to 24 do a[n]:=6*a[n-1]+4*a[n-2] -a[n-3] od: seq(a[n],n=0..24);
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Mathematica
LinearRecurrence[{6,4,-1}, {1,5,34}, 30] (* G. C. Greubel, Oct 27 2019 *) -
PARI
Vec((1 - x) / (1 - 6*x - 4*x^2 + x^3) + O(x^30)) \\ Colin Barker, Jun 06 2017
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Sage
def A102436_list(prec): P.
= PowerSeriesRing(ZZ, prec) return P((1-x)/(1-6*x-4*x^2+x^3)).list() A102436_list(30) # G. C. Greubel, Oct 27 2019
Formula
a(n) = 6*a(n-1) + 4*a(n-2) - a(n-3) for n>=3.
G.f.: (1-x) / (1-6*x-4*x^2+x^3).
Comments