A028468 Number of perfect matchings in graph P_{6} X P_{n}.

1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479

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, Counting Hamiltonian cycles in product graphs

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.

Thotsaporn ”Aek” Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.

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<x>:=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

N. J. A. Sloane, Per H. Lundow

Status

approved