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 A033507 Number of matchings in graph P_{4} X P_{n}. 6
 1, 5, 71, 823, 10012, 120465, 1453535, 17525619, 211351945, 2548684656, 30734932553, 370635224561, 4469527322891, 53898461609719, 649966808093412, 7838012982224913, 94519361817920403, 1139818186429110279, 13745178487929574337, 165754445655292452448 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 David Friedhelm Kind, The Gunport Problem: An Evolutionary Approach, De Montfort University (Leicester, UK, 2020). Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden. Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998. Index entries for linear recurrences with constant coefficients, signature (9,41,-41,-111,91,29,-23,-1,1). FORMULA From Sergey Perepechko, Apr 24 2013: (Start) 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) EXAMPLE 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 MAPLE 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); # Sergey Perepechko, Apr 24 2013 MATHEMATICA 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 *) PROG (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:=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) def A033507_list(prec): P. = 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() A033507_list(30) # G. C. Greubel, Oct 26 2019 (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 CROSSREFS Column 4 of triangle A210662. Row sums of A100265. For perfect matchings see A005178. Cf. A033508-A033511. Bisection (even part) gives A260034. Sequence in context: A197427 A197668 A064752 * A092250 A362159 A371326 Adjacent sequences: A033504 A033505 A033506 * A033508 A033509 A033510 KEYWORD nonn AUTHOR Per H. Lundow EXTENSIONS Edited by N. J. A. Sloane, Nov 15 2009 STATUS approved

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Last modified April 20 17:12 EDT 2024. Contains 371845 sequences. (Running on oeis4.)