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%I #21 Nov 05 2020 06:11:54
%S 1,3,32,229,1845,14320,112485,880163,6895792,54003765,422983905,
%T 3312866080,25947198337,203223953179,1591695681488,12466511517581,
%U 97640484615909,764741896529104,5989627994067061,46912093390144139,367425909133064576,2877761124002870925
%N Number of perfect matchings in graph P_{2} X P_{3} X P_{n}.
%C lim(a(n)^(1/n), n -> infinity) = 7.832221... - _Emeric Deutsch_, Oct 14 2006
%H Per Hakan Lundow, <a href="http://www.theophys.kth.se/~phl/Text/1factors.pdf">Computation of matching polynomials and the number of 1-factors in polygraphs</a>, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
%H Per Hakan Lundow, <a href="http://www.theophys.kth.se/~phl/Text/1factors2.ps.gz">Enumeration of matchings in polygraphs</a>, 1998.
%H R. J. Mathar, <a href="https://arxiv.org/abs/1406.7788">Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices</a>, arXiv:1406.7788 (2014), eq. (37).
%H H. Narumi and H. Hosoya, <a href="http://dx.doi.org/10.1063/1.530236">Generalized expression of the perfect matching number of 2 X 3 X n lattices</a>, J. Math. Phys. 34 (3), 1993, 1043-1051.
%F 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
%F 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
%p 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
%Y Column k=3 of A181206.
%K nonn
%O 0,2
%A _Per H. Lundow_
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