A102436 - OEIS

%I #19 Sep 08 2022 08:45:16

%S 1,5,34,223,1469,9672,63685,419329,2761042,18179883,119704137,

%T 788183312,5189736537,34171448333,224999452834,1481492773799,

%U 9754783005797,64229669677144,422915657312253,2784657839576297,18335379997029650

%N 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'.

%C Row sums of A102435. The number of matchings of the ladder graph L(n)=P_2 X P_n is given in A030186.

%C Number of tilings of a 2xn board with squares of 2 colors and dominoes of 1 color [Katz-Stenson]. - _R. J. Mathar_, Apr 17 2009

%H Colin Barker, <a href="/A102436/b102436.txt">Table of n, a(n) for n = 0..1000</a>

%H M. Katz, C. Stenson, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Stenson/stenson8.html">Tiling a 2xn-board with squares and dominoes</a>, J. Int. Seq. 12 (2009) # 09.2.2

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,4,-1).

%F a(n) = 6*a(n-1) + 4*a(n-2) - a(n-3) for n>=3.

%F G.f.: (1-x) / (1-6*x-4*x^2+x^3).

%e 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}).

%p 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);

%t LinearRecurrence[{6,4,-1}, {1,5,34}, 30] (* _G. C. Greubel_, Oct 27 2019 *)

%o (PARI) Vec((1 - x) / (1 - 6*x - 4*x^2 + x^3) + O(x^30)) \\ _Colin Barker_, Jun 06 2017

%o (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

%o (Sage)

%o def A102436_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P((1-x)/(1-6*x-4*x^2+x^3)).list()

%o A102436_list(30) # _G. C. Greubel_, Oct 27 2019

%o (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

%Y Cf. A030186, A102435, A253265.

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Jan 08 2005