A001409 Number of 2n-step polygons on cubic lattice.

1, 0, 3, 22, 207, 2412, 31754, 452640, 6840774, 108088232, 1768560270, 29764630632, 512705615350, 9005206632672, 160810554015408, 2912940755956084, 53424552150523386

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• more

Examples

			From _M. F. Hasler_, Jun 17 2025: (Start)
For n = 2, the three 4-step polygons are the 1 X 1 squares orthogonal to one of the three coordinate axes. (The sequence counts the polygons up to translations.)
For n = 3, the 22 six-step polygons can be partitioned into:
  - six 2 X 1 rectangles (two in each of the previously considered planes);
  - twelve L- or "seat" shaped polygons (as one can get by gluing together two 1 X 1 squares in a 90 degree angle along one side, or by folding a 2 X 1 rectangle by 90 degrees along the common side of its 1 X 1 square halves): choose one of the six half axes for the orientation of one of the squares, and one of the four orthogonal axes for the other, then divide by two because the order of the two choices doesn't matter;
  - four polygons obtained by making three steps in direction of distinct axes (e.g., in direction of the three unit vectors) and then the same three steps in the opposite direction. The four inequivalent instances are obtained by rotating one of them three times by 90° around the same fixed axis. (End)
		

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor., 40 (2007), pp. 10973-11017, Table A5.
• G. S. Rushbrooke and J. Eve, High-temperature Ising partition function and related noncrossing polygons for the simple cubic lattice, J. Math. Physics, 3 (1962), pp. 185-189.

Crossrefs

Cf. A001412, A001413.

Extensions

More terms from R. J. Mathar, Aug 31 2007