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%I #26 Feb 10 2020 12:33:04
%S 1,8,20,62,132,336,688,1578,3190,6902,13878,29038,58238,119518,239390,
%T 485822,972414,1960830,3923326,7882494,15768574,31616510,63240702,
%U 126655486,253327358,507033598,1014102014,2029023230,4058120190,8118001662,16236158974,32476086270
%N Number of Hamiltonian paths in P_3 X P_n.
%D F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
%H Andrew Howroyd, <a href="/A003685/b003685.txt">Table of n, a(n) for n = 1..500</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cpaper.zip">On the number of specific spanning subgraphs of the graphs G X P_n</a>, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
%H F. Faase, <a href="http://www.iwriteiam.nl/counting.html">Counting Hamiltonian cycles in product graphs</a>
%H F. Faase, <a href="http://www.iwriteiam.nl/Cresults.html">Results from the counting program</a>
%H A. Kloczkowski, and R. L. Jernigan, <a href="https://doi.org/10.1063/1.477128">Transfer matrix method for enumeration and generation of compact self-avoiding walks. I. Square lattices</a>, The Journal of Chemical Physics 109, 5134 (1998); doi: 10.1063/1.477128.
%F a(n) = 3*a(n-1) + 2*a(n-2) - 12*a(n-3) + 4*a(n-4) + 12*a(n-5) - 8*a(n-6), n>8.
%F From _David Bevan_, Jul 21 2006: (Start)
%F a(2*m) = 121*2^(2*m-4) - 4*m*2^m - 25*2^(m-2) - 2, m > 1.
%F a(2*m+1) = 121*2^(2*m-3) - 31*m*2^(m-2) - 23*2^(m-1) - 2, m > 0.
%F a(n) = 8*a(n-2) - 20*a(n-4) + 16*a(n-6) + 6, n > 8. (End)
%F O.g.f.: (2*x^7-8*x^6+12*x^5-2*x^4-2*x^3-6*x^2+5*x+1)*x/((2*x-1)*(-1+2*x^2)^2*(-1+x)). - _R. J. Mathar_, Dec 05 2007
%Y Row n=3 of A332307.
%K nonn
%O 1,2
%A _Frans J. Faase_
%E Terms a(29) and beyond from _Andrew Howroyd_, Feb 10 2020
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