A392389 Number of free polybends with n cells.

1, 2, 3, 6, 8, 15, 25, 45, 74, 135, 233, 414, 715, 1279, 2248, 3952, 6924, 12279, 21623, 37895, 66551, 117379, 206576, 361376, 634301, 1115258, 1959444, 3424197, 6003124, 10532541, 18475802, 32262960, 56494834, 98969405, 173373136, 302571376, 529281709

Offset

1,2

Comments

The cells of a polybend are quarter annuli with a fixed outer radius R and inner radius 2-R, where 1 < R < sqrt(2), with the two straight parts of their boundaries parallel to the coordinate axes. Cells are connected along their straight edges. Overlapping cells are not permitted.

Alternatively, the cells can be taken to be isosceles right triangles (like polyaboloes) which can be glued along their short sides. Cells are allowed to meet along their long sides, but the polybends must be connected through short side connections only.

Cells can be represented as integer points in the plane, with the neighbors of (x,y) being (x,y-s), (x-r,y), (x+r,y-s), (x-r,y+s), where r = (-1)^(x mod 2) and s = (-1)^(y mod 2). If (x,y) is a cell, the cells (x+r,y) and (x,y+s) are not permitted in the same polybend.

Links

Table of n, a(n) for n=1..37.

Pontus von Brömssen, Illustration for n=1..6.

Kate Jones, A Periodic Table of polyform puzzles, 2020. (See under "polybends".)

Kate Jones, Essential Polyforms. (See under "ChooChooLoops".)

Crossrefs

Cf. A390885, A390993, A392390 (one-sided), A392391 (fixed).

Cf. closed polybends: A056841 (4n cells, smallest possible enclosed area), A333233 (by enclosed area).

Sequence in context: A048809 A300671 A268645 * A047001 A174021 A267007

Adjacent sequences: A392386 A392387 A392388 * A392390 A392391 A392392

Keyword

nonn

Author

Pontus von Brömssen, Jan 10 2026

Extensions

a(28)-a(37) from Christian Sievers, Jan 13 2026

Status

approved