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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'. 5
1, 5, 34, 223, 1469, 9672, 63685, 419329, 2761042, 18179883, 119704137, 788183312, 5189736537, 34171448333, 224999452834, 1481492773799, 9754783005797, 64229669677144, 422915657312253, 2784657839576297, 18335379997029650 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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).

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).

EXAMPLE

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}).

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);

MATHEMATICA

LinearRecurrence[{6, 4, -1}, {1, 5, 34}, 30] (* G. C. Greubel, Oct 27 2019 *)

PROG

(PARI) Vec((1 - x) / (1 - 6*x - 4*x^2 + x^3) + O(x^30)) \\ Colin Barker, Jun 06 2017

(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

(Sage)

def A102436_list(prec):

    P.<x> = 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

(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

CROSSREFS

Cf. A030186, A102435, A253265.

Sequence in context: A127816 A024063 A015545 * A291027 A033889 A120469

Adjacent sequences:  A102433 A102434 A102435 * A102437 A102438 A102439

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Jan 08 2005

STATUS

approved

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Last modified February 26 11:03 EST 2020. Contains 332279 sequences. (Running on oeis4.)