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A033507
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Number of matchings in graph P_{4} X P_{n}.
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6
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1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448
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OFFSET
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0,2
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REFERENCES
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H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Phys., 26(1985), 157-167.
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LINKS
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FORMULA
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a(n) = 9*a(n-1) +41*a(n-2) -41*a(n-3) -111*a(n-4) +91*a(n-5) +29*a(n-6) -23*a(n-7) -a(n-8) +a(n-9).
G.f.: (1-x) * (1 -3*x -18*x^2 +2*x^3 +12*x^4 +x^5 -x^6) / (1 -9*x -41*x^2 +41*x^3 +111*x^4 -91*x^5 -29*x^6 +23*x^7 +x^8 -x^9). (End)
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EXAMPLE
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a(1) = 5: the graph is
. o-o-o-o
and the five matchings are
. o o o o
. o-o o o
. o o-o o
. o o o-o
. o-o o-o
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MAPLE
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a:=array(0..20, [1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945]):
for j from 9 to 20 do
a[j]:=9*a[j-1]+41*a[j-2]-41*a[j-3]-111*a[j-4]+91*a[j-5]+
29*a[j-6]-23*a[j-7]-a[j-8]+a[j-9]
od:
convert(a, list);
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MATHEMATICA
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LinearRecurrence[{9, 41, -41, -111, 91, 29, -23, -1, 1}, {1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945}, 30] (* Harvey P. Dale, Mar 27 2015 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9)) \\ G. C. Greubel, Oct 26 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) )); // G. C. Greubel, Oct 26 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x)*(1 -3*x-18*x^2+2*x^3+12*x^4+x^5-x^6)/(1-9*x-41*x^2+41*x^3+111*x^4-91*x^5 -29*x^6+23*x^7+x^8-x^9) ).list()
(GAP) a:=[1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945];; for n in [10..30] do a[n]:=9*a[n-1]+41*a[n-2]-41*a[n-3]-111*a[n-4]+91*a[n-5] +29*a[n-6]-23*a[n-7]-a[n-8]+a[n-9]; od; a; # G. C. Greubel, Oct 26 2019
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CROSSREFS
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Bisection (even part) gives A260034.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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