A003685 Number of Hamiltonian paths in P_3 X P_n.

1, 8, 20, 62, 132, 336, 688, 1578, 3190, 6902, 13878, 29038, 58238, 119518, 239390, 485822, 972414, 1960830, 3923326, 7882494, 15768574, 31616510, 63240702, 126655486, 253327358, 507033598, 1014102014, 2029023230, 4058120190, 8118001662, 16236158974, 32476086270

Offset: 1

Author

Frans J. Faase

Keywords

• nonn

References

• F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..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
• A. Kloczkowski, and R. L. Jernigan, Transfer matrix method for enumeration and generation of compact self-avoiding walks. I. Square lattices, The Journal of Chemical Physics 109, 5134 (1998); doi: 10.1063/1.477128.

Crossrefs

Row n=3 of A332307.

Formula

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.

From David Bevan, Jul 21 2006: (Start)

a(2*m) = 121*2^(2*m-4) - 4*m*2^m - 25*2^(m-2) - 2, m > 1.

a(2*m+1) = 121*2^(2*m-3) - 31*m*2^(m-2) - 23*2^(m-1) - 2, m > 0.

a(n) = 8*a(n-2) - 20*a(n-4) + 16*a(n-6) + 6, n > 8. (End)

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

Extensions

Terms a(29) and beyond from Andrew Howroyd, Feb 10 2020