A028469 Number of perfect matchings in graph P_{7} X P_{2n}.
1, 21, 781, 31529, 1292697, 53175517, 2188978117, 90124167441, 3710708201969, 152783289861989, 6290652543875133, 259009513044645817, 10664383939345916681, 439092316687230373293, 18079062471131097321077
Offset: 0
Keywords
References
- F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..500
- 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
- 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], Table 6.
- Index entries for linear recurrences with constant coefficients, signature (56, -672, 2632, -4094, 2632, -672, 56, -1).
Crossrefs
Row 7 of array A099390.
Programs
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Mathematica
a[n_] := Product[2(2+Cos[k Pi/4]+Cos[2j Pi/(2n+1)]), {k, 1, 3}, {j, 1, n}] // Round; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Aug 20 2018 *)
Formula
G.f.: (-x^7 +35*x^6 -277*x^5 +727*x^4 -727*x^3 +277*x^2 -35*x +1) / (x^8 -56*x^7 +672*x^6 -2632*x^5 +4094*x^4 -2632*x^3 +672*x^2 -56*x +1).
(Faase:) If b(n) denotes the number of perfect matchings in P_7 X P_n we have:
b(1) = 0,
b(2) = 21,
b(3) = 0,
b(4) = 781,
b(5) = 0,
b(6) = 31529,
b(7) = 0,
b(8) = 1292697,
b(9) = 0,
b(10) = 53175517,
b(11) = 0,
b(12) = 2188978117,
b(13) = 0,
b(14) = 90124167441,
b(15) = 0,
b(16) = 3710708201969, and
b(n) = 56b(n-2) - 672b(n-4) + 2632b(n-6) - 4094b(n-8) + 2632b(n-10) - 672b(n-12) + 56b(n-14) - b(n-16).
Extensions
Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009