cp's OEIS Frontend

A350148 Number of distinct (left- or right-handed, but not both) two-dimensional, Hilbert-style space-filling curve motifs on the 2n+1 X 2n+1 square subdivision, that, when recursively iterated using strict edge-replacement, create always self-avoiding paths formed of sub-square edges in the lattice.

%I A350148 #24 Jul 25 2023 10:12:57
%S A350148 1,0,1,7,10101,20305328
%N A350148 Number of distinct (left- or right-handed, but not both) two-dimensional, Hilbert-style space-filling curve motifs on the 2n+1 X 2n+1 square subdivision, that, when recursively iterated using strict edge-replacement, create always self-avoiding paths formed of sub-square edges in the lattice.
%C A350148 The paper proves that all motifs for a given n>=0 fall into F(n-1) zipping modes, where F(n) is the n-th Fibonacci number.  Each mode represents a fixed state of all edges along the boundary of the motif that allows it to zip with itself.  For n=4, 10101 = 600 + 9441 (F(4-1) = 2 modes); For n=5, 20305328 = 58936 + 19854452 + 391940 (F(5-1) = 3 modes).
%C A350148 A000532 represent Hilbert-style motifs also, but they are self-avoiding paths connecting sub-square centers.  This sequence counts Hilbert-style motifs as self-avoiding paths along sub-square edges.  In both cases, these self-avoiding paths in the square lattice can be considered Hamiltonian cycles on a 2D toroidal grid-graph.
%H A350148 Douglas M. McKenna, <a href="http://ecajournal.haifa.ac.il/Volume2022/ECA2022_S2A13.pdf">Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-style Square-filling Curves</a>, Enumerative Combinatorics and Applications, 2:2 #S2R13 (2021).
%H A350148 Douglas M. McKenna, <a href="https://archive.bridgesmathart.org/2023/bridges2023-91.html">Are Maximally Unbalanced Hilbert-Style Square-Filling Curve Motifs a Drawing Medium?</a>, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 91-98.
%e A350148 The n=0 case is the trivial/idempotent identity motif and does not converge to a space-filling curve.  There are no solutions for the 2n X 2n case.
%Y A350148 Cf. A000532.
%K A350148 nonn,walk,more
%O A350148 0,4
%A A350148 _Douglas M. McKenna_, Dec 16 2021