A028447 Number of perfect matchings in graph P_{2} X P_{3} X P_{n}.

1, 3, 32, 229, 1845, 14320, 112485, 880163, 6895792, 54003765, 422983905, 3312866080, 25947198337, 203223953179, 1591695681488, 12466511517581, 97640484615909, 764741896529104, 5989627994067061, 46912093390144139, 367425909133064576, 2877761124002870925

Offset: 0

Author

Per H. Lundow

Keywords

• nonn

Comments

lim(a(n)^(1/n), n -> infinity) = 7.832221... - Emeric Deutsch, Oct 14 2006

Links

• 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, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 (2014), eq. (37).
• H. Narumi and H. Hosoya, Generalized expression of the perfect matching number of 2 X 3 X n lattices, J. Math. Phys. 34 (3), 1993, 1043-1051.

Crossrefs

Column k=3 of A181206.

Programs

Maple

a[0]:=1: a[1]:=3: a[2]:=32: a[3]:=229: a[4]:=1845: a[5]:=14320: a[6]:=112485: a[7]:=880163: a[8]:=6895792: a[9]:=54003765: a[10]:=422983905: for n from 11 to 20 do a[n]:=6*a[n-1]+21*a[n-2]-42*a[n-3]-89*a[n-4]+68*a[n-5]+89*a[n-6]-42*a[n-7]-21*a[n-8]+6*a[n-9]+a[n-10] od: seq(a[n], n=0..30); # Emeric Deutsch, Oct 14 2006

Formula

a(n) = 6a(n - 1) + 21a(n - 2) - 42a(n - 3) - 89a(n - 4) + 68a(n - 5) + 89a(n - 6) - 42a(n - 7) - 21a(n - 8) + 6a(n - 9) + a(n - 10). - Emeric Deutsch, Oct 14 2006

G.f.: ( -1 +3*x +7*x^2 -16*x^3 -14*x^4 +16*x^5 +7*x^6 -3*x^7 -x^8 ) / ( (x^2-x-1) *(x^8 +7*x^7 -13*x^6 -48*x^5 +28*x^4 +48*x^3 -13*x^2 -7*x+1) ). - R. J. Mathar, Dec 06 2013