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 A028468 Number of perfect matchings in graph P_{6} X P_{n}. 7
 1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 REFERENCES F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154. R. P. Stanley, Enumerative Combinatorics I, p. 292. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450 F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154. F. Faase, Results from the counting program David Klarner, Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52 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. R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 5. Index entries for linear recurrences with constant coefficients, signature (1,20,10,-38,-10,20,-1,-1). FORMULA From  N. J. A. Sloane, Feb 03 2009: (Start) a(1) = 1, a(2) = 13, a(3) = 41, a(4) = 281, a(5) = 1183, a(6) = 6728, a(7) = 31529, a(8) = 167089, a(9) = 817991, a(10) = 4213133, a(11) = 21001799, a(12) = 106912793, a(13) = 536948224, a(14) = 2720246633, and a(n) = 40*a(n-2) - 416*a(n-4) + 1224*a(n-6) - 1224*a(n-8) + 416*a(n-10) - 40*a(n-12) + a(n-14). (From Faase's web page.) (End) G.f.: (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1) / ( (1-x) *(1+x) *(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1) ). a(n) = a(n-1)+20*a(n-2)+10*a(n-3)-38*a(n-4)-10*a(n-5)+20*a(n-6)-a(n-7)-a(n-8). - Sergey Perepechko, Sep 23 2018 MAPLE seq(coeff(series((1+2*x-x^2)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(x+1)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1)), x, n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 23 2018 MATHEMATICA a[n_] := Product[2(2 + Cos[(2 k Pi)/7] + Cos[(2 j Pi)/(n+1)]), {k, 1, 3}, {j, 1, n/2}] // Round; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 19 2018, after A099390 *) LinearRecurrence[{1, 20, 10, -38, -10, 20, -1, -1}, {1, 1, 13, 41, 281, 1183, 6728, 31529}, 30] (* Vincenzo Librandi, Nov 24 2018 *) PROG (PARI) my(x='x+O('x^30)); Vec(-(x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(1+x)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1))) \\ Altug Alkan, Mar 23 2016 (MAGMA) m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1)*(x^3+ 6*x^2+5*x+1)) )); // G. C. Greubel, Nov 25 2018 (Sage) s=((x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1))).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 25 2018 CROSSREFS Row 6 of arrays A099390, A189006. Column k=2 of A251072. Cf. A005178. Sequence in context: A231885 A167240 A147247 * A146995 A102130 A080186 Adjacent sequences:  A028465 A028466 A028467 * A028469 A028470 A028471 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 19 11:09 EDT 2019. Contains 328216 sequences. (Running on oeis4.)