cp's OEIS Frontend

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

%I A001409 M3095 N1254 #33 Jun 19 2025 00:16:11
%S A001409 1,0,3,22,207,2412,31754,452640,6840774,108088232,1768560270,
%T A001409 29764630632,512705615350,9005206632672,160810554015408,
%U A001409 2912940755956084,53424552150523386
%N A001409 Number of 2n-step polygons on cubic lattice.
%D A001409 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001409 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001409 N. Clisby, R. Liang, and G. Slade, <a href="http://dx.doi.org/10.1088/1751-8113/40/36/003">Self-avoiding walk enumeration via the lace expansion</a>, J. Phys. A: Math. Theor., 40 (2007), pp. 10973-11017, Table A5.
%H A001409 G. S. Rushbrooke and J. Eve, <a href="https://doi.org/10.1063/1.1703777">High-temperature Ising partition function and related noncrossing polygons for the simple cubic lattice</a>, J. Math. Physics, 3 (1962), pp. 185-189.
%e A001409 From _M. F. Hasler_, Jun 17 2025: (Start)
%e A001409 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.)
%e A001409 For n = 3, the 22 six-step polygons can be partitioned into:
%e A001409   - six 2 X 1 rectangles (two in each of the previously considered planes);
%e A001409   - 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;
%e A001409   - 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)
%Y A001409 Cf. A001412, A001413.
%K A001409 nonn,walk,more
%O A001409 0,3
%A A001409 _N. J. A. Sloane_
%E A001409 More terms from _R. J. Mathar_, Aug 31 2007