A394184 Number of primitive tessellations of paths of length n in the square lattice, reduced for symmetry (with rotations, reflections, and translations not counted as distinct).

1, 1, 1, 4, 4, 9, 11, 24, 32, 56, 81, 169, 230, 431, 740, 1360, 2242, 4357, 7553, 14611, 26691, 50262, 94037, 182581, 342243, 656846, 1259106

Offset

1,4

Comments

Tessellations are formed by adjoining successive endpoints of a repeating path of length n (distinct paths tallied in A037245(n)) to form an endless self-avoiding path, and then repeating this endless path to form a regular tessellation (tiling lattices tallied in in A000203(n), or A145393(n) allowing for rotations and reflections), with each repeating step using only translations and not rotations or reflections.

Links

Table of n, a(n) for n=1..27.

Charles L. Hohn, Illustrations of a(1)-a(8) plus A393645 (unreduced), 2026.

Example

a(1) = 1: a 1-unit line segment, repeated into an endless straight line, then repeated into an endless field of straight stripes.
a(2) = 1: a path of 2 1-unit segments at a right angle, repeated into an endless right angle zigzag path, then repeated into an endless field of zigzag stripes.
a(6) = 9 (shown here as the paths that can form tessellations, with uppercase letters denoting positive direction steps in XY coordinates and lowercase denoting negative steps): XXXXXY, XXXXYY, XXXYXY, XXXYYY, XXXYxY, XXYXYY, XYXYXy, and XXYxxY (produces 2 distinct regular tessellations so counts as 2).
Examples of paths not counted for n = 6: XXYXXY (nonprimitive, counted as XXY for n = 3), XXXXYX (endless path is congruent to that of XXXXXY), XXXXYx: (endless path is self-intersecting), and XXXYXy (endless path does not tessellate).
XXXXYY and XXXYYY are counted because even though they are not primitive paths (reducing to XXY and XY respectively), their tessellations are not congruent to those of their corresponding reduced paths.

Crossrefs

Cf. A393645 (unreduced), A067769 (tilings of polyominoes), A037245, A000203, A145393.

Sequence in context: A116682 A168157 A222045 * A088190 A378032 A092322

Adjacent sequences: A394181 A394182 A394183 * A394185 A394186 A394187

Keyword

nonn,walk,more,hard

Author

Charles L. Hohn, May 04 2026

Status

approved