Enumeration of polycube snakes

Question

Has anyone enumerated polycube snakes formed of $n$ cubes? A polycube is an object created by gluing cubes face-to-face. A polycube snake's dual graph is a path. I would be especially interested if the snake's surface forms a manifold, i.e., there are not edge-edge and vertex-vertex nonmanifold touchings. The closest I've found is this, which analyzes a particular class ("partially directed") of snakes:

Goupil, Alain, Marie-Eve Pellerin, and Jérôme de Wouters d’Oplinter. "Partially directed snake polyominoes." Discrete Applied Mathematics 236 (2018): 223-234. arXiv abs

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          ^Fig.7b

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Answer 1

How about OEIS A000162 Number of 3-dimensional polyominoes?

--- rk

Answer 2

This sequence doesn't seem to be in the OEIS yet! Perhaps someone with an account would like to add it.

Imagine standing at one end of the tunnel formed by a snake, and walking from one end to the other, recording each turn you make as you step from one cube to another; this maps the snake's shape to a string of $n-1$ letters (Straight, Left, Right, Up, Down). By standing initially facing from the first cube into the second, and with an appropriate rotation, you can always arrange it so that the first letter is S and the first non-S turn is a Right turn.

I wrote a little C++ program to enumerate unique 3D non-ouroboros polycube snakes, both the "directed" version (where the snake's head is distinguishable from its tail) and the "undirected" version (where it's not). I'm not 100% sure that my program is correct, but the results seem plausible.

We might follow Gardner/Kim and consider SRR (the square-tetromino) or SSRSRSR (the O-octomino) to be ouroboros polycube snakes, whose head and tail are adjacent. (The 2x3 rectangle SRRSR is not a snake at all, because its head has 3 neighbors.) But an ouroboros can be represented by as many different letter-strings as it has cubes; e.g. SSSSRSRSLRSRS is not only the same as SSLSLRSLSLSS but also the same as SSSRSRSLRSRSS and SSRSRSLRSRSSR and so on. I've manually verified my program's enumeration of the 13 ten-cube ouroboroi.

Cubes	Directed non-ouroboroi	Undirected non-ouroboroi	Undirected ouroboroi
1	1	1
2	1	1	0
3	2: SS SR	2
4	6: SSS SSR SRS SRL SRU SRD	5: SSS SRS SRL SRU SRD	1: SRR
5	23	16: SSSS SSRS SRSS SRSR SRSL SRSU SRSD SRLS SRLR SRLD SRUS SRUL SRDS SRDR SRDL SRDU
6	93	54	1: SRURU
7	386	212
8	1590	827	3: SRSRSRS SRDRSRD SRDRSDR
9	6583	3369
10	27046	13653	13
11	111460	56052
12	456937	229004	122
13	1877277	939935
14	7683372	3843859	1115
15	31497124	15753903
16	128743825	64380796	12562
17	526907643	263475472
18	2151488689	1075780425	147350
19	8794145222	4397161320
20	35878493709	17939394036	1810165
21	146503034913	73251877235
22	597291499318	298646347226	22812552
23	2436903747928	1218453344740

OEIS A???: 1 1 2 6 23 93 386 1590 6583 27046 111460 456937 1877277 7683372 31497124 128743825 526907643 2151488689 8794145222 35878493709 146503034913 597291499318 2436903747928 ...

OEIS A???: 1 1 2 5 16 54 212 827 3369 13653 56052 229004 939935 3843859 15753903 64380796 263475472 1075780425 4397161320 17939394036 73251877235 298646347226 1218453344740 ...

OEIS A???: 1 1 2 6 16 55 212 830 3369 13666 56052 229126 939935 3844974 15753903 64393358 263475472 1075927775 4397161320 17941204201 73251877235 298669159778 1218453344740 ...