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.

1, 0, 1, 7, 10101, 20305328

Offset

0,4

Comments

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).

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.

Links

Table of n, a(n) for n=0..5.

Douglas M. McKenna, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-style Square-filling Curves, Enumerative Combinatorics and Applications, 2:2 #S2R13 (2021).

Douglas M. McKenna, Are Maximally Unbalanced Hilbert-Style Square-Filling Curve Motifs a Drawing Medium?, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 91-98.

Example

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.

Crossrefs

Cf. A000532.

Sequence in context: A074489 A067248 A152007 * A242773 A333338 A131676

Adjacent sequences: A350145 A350146 A350147 * A350149 A350150 A350151

Keyword

nonn,walk,more

Author

Douglas M. McKenna, Dec 16 2021

Status

approved