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A384173 Number of Hamiltonian paths from NW to SW corners in an n X n grid reduced for symmetry, i.e., where reflection about the x-axis is not counted as distinct.

%I A384173 #17 Jul 23 2025 10:03:59
%S A384173 1,1,1,5,43,897,44209,4467927,1043906917,506673590576,555799435739334,
%T A384173 1284472450789974196,6625529679919810063544,
%U A384173 72597408139909172033687226,1762085630816152820582838187465,91326629994353561722347679614188407
%N A384173 Number of Hamiltonian paths from NW to SW corners in an n X n grid reduced for symmetry, i.e., where reflection about the x-axis is not counted as distinct.
%C A384173 When n is odd there are no symmetric Hamiltonian paths from NW to SW corners, and therefore a(n) = A000532(n)/2.
%D A384173 J. L. Jacobsen, Exact enumeration of Hamiltonian circuits, walks and chains in two and three dimensions, J. Phys. A: Math. Theor. 40 (2007) 14667-14678.
%D A384173 J.-M. Mayer, C. Guez and J. Dayantis, Exact computer enumeration of the number of Hamiltonian paths in small square plane lattices, Physical Review B, Vol. 42 Number 1, 1990.
%H A384173 Oliver R. Bellwood, Heitor P. Casagrande, and William J. Munro, <a href="https://arxiv.org/abs/2507.11820">Fractal Path Strategies for Efficient 2D DMRG Simulations</a>, arXiv:2507.11820 [cond-mat.str-el], 2025. See p. 4.
%F A384173 a(n) = A000532(n)/2 for odd n.
%e A384173 The two paths of A000532(3) = 2 are equivalent under reflection about the x-axis:
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%Y A384173 Cf. A265914, A209077.
%K A384173 nonn,walk
%O A384173 1,4
%A A384173 _Oliver R. Bellwood_, May 21 2025