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A033506
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Number of matchings in graph P_{3} X P_{n}.
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4
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1, 3, 22, 131, 823, 5096, 31687, 196785, 1222550, 7594361, 47177097, 293066688, 1820552297, 11309395995, 70254767718, 436427542283, 2711118571311, 16841658983944, 104621568809247, 649916534985369, 4037327172325542
(list;
graph;
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history;
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internal format)
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OFFSET
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0,2
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 50, 999.
Per Hakan Lundow, "Computation of matching polynomials and the number of 1-factors in polygraphs", Research reports, No 12, 1996, Department of Mathematics, Umea University.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Svenja Huntemann, Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020).
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. C. Read, The dimer problem for narrow rectangular arrays: A unified method of solution, and some extensions, Aequationes Mathematicae, 24 (1982), 47-65.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Index entries for linear recurrences with constant coefficients, signature (4,14,0,-10,0,1).
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FORMULA
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G.f.: (1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)). - Sergey Perepechko, Apr 19 2013
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MAPLE
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seq(coeff(series((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5 )), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 26 2019
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MATHEMATICA
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CoefficientList[Series[(1-2x-x^2)(1+x-x^2)/((1+x)(1-5x-9x^2+9x^3+x^4-x^5) ), {x, 0, 30}], x] (* Harvey P. Dale, Dec 05 2014 *)
LinearRecurrence[{4, 14, 0, -10, 0, 1}, {1, 3, 22, 131, 823, 5096}, 30] (* Harvey P. Dale, Dec 05 2014 *)
Table[RootSum[-1 +# +9#^2 -9#^3 -5#^4 +#^5 &, 1436541#^n + 3941068#^(n+1) -6086452#^(n+2) -2800519#^(n+3) +591744#^(n+4) &]/10204570 -(-1)^n/5, {n, 20}] (* Eric W. Weisstein, Oct 02 2017 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2 +9*x^3+x^4-x^5))) \\ G. C. Greubel, Oct 26 2019
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)) )); // G. C. Greubel, Oct 26 2019
(Sage)
def A033506_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-2*x-x^2)*(1+x-x^2)/((1+x)*(1-5*x-9*x^2+9*x^3+x^4-x^5)) ).list()
A033506_list(30) # G. C. Greubel, Oct 26 2019
(GAP) a:=[1, 3, 22, 131, 823, 5096];; for n in [7..30] do a[n]:=4*a[n-1] +14*a[n-2]-10*a[n-4]+a[n-6]; od; a; # G. C. Greubel, Oct 26 2019
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CROSSREFS
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Column 3 of triangle A210662. Row sums of A100245.
Cf. A030186, A033507, A033508, A033509, A033510, A033511.
Sequence in context: A006283 A232017 A100511 * A091639 A091636 A321003
Adjacent sequences: A033503 A033504 A033505 * A033507 A033508 A033509
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KEYWORD
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nonn
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AUTHOR
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Per H. Lundow
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STATUS
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approved
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