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A363577 Number of inequivalent Hamiltonian paths starting in the lower left corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent).

%I A363577 #27 Jul 23 2025 10:04:15
%S A363577 1,1,3,23,347,10199,683227,85612967,25777385143,14396323278040,
%T A363577 19799561204761862,50351228336401026361,319210377672595552740369,
%U A363577 3736517399241599771428011100,109790442395888863208285555153329,5952238893391106787883489313797219949
%N A363577 Number of inequivalent Hamiltonian paths starting in the lower left corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent).
%C A363577 Equivalently, number of inequivalent Hamiltonian paths starting in a corner of an n X n grid graph (paths differing only by rotations and reflections are regarded as equivalent). - _Martin Ehrenstein_, Jul 08 2023
%H A363577 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.
%H A363577 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A363577 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a>
%H A363577 <a href="/index/Gra#graphs">Index entries for sequences related to graphs, Hamiltonian</a>
%e A363577 There are 3 paths for n=3:
%e A363577   +--+--+    +--+--+    +--+  +
%e A363577   |     |    |     |    |  |  |
%e A363577   +  +  +    +  +--+    +  +  +
%e A363577   |  |  |    |  |       |  |  |
%e A363577   +  +--+    +  +--+    +  +--+
%e A363577 A fourth path:
%e A363577   +--+--+
%e A363577         |
%e A363577   +--+  +
%e A363577   |  |  |
%e A363577   +  +--+
%e A363577 is the same as the second one in the row above after a 90-degree rotation.
%e A363577 All paths starting E are the same as the corresponding ones starting N after reflection in the forward diagonal.
%Y A363577 Cf. A000532, A001184, A003763, A007764, A120443, A121785, A121789, A145157, A271507, A384173.
%K A363577 nonn,hard,walk
%O A363577 1,3
%A A363577 _Lars Blomberg_, Jun 10 2023
%E A363577 a(1) added by _N. J. A. Sloane_, Jun 10 2023
%E A363577 a(8)-a(9) from _Martin Ehrenstein_, Jul 08 2023
%E A363577 a(10)-a(16) from _Oliver R. Bellwood_, Jun 06 2025