A001997 Number of different shapes formed by bending a piece of wire of length n in the plane.

1, 1, 2, 4, 10, 24, 66, 176, 493, 1362, 3821, 10660, 29864, 83329, 232702, 648182, 1804901, 5015725, 13931755, 38635673, 107090666, 296449133, 820271143, 2267225157, 6264244414, 17291930470

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• more
• nice
• walk

Comments

The wire is marked into n equal segments by n-1 marks, is bent at right angles at each of one or more of these points, making each segment parallel to one of two rectangular axes. (Stays in plane, bends are of 0 or +-90 degs.) May cross itself but is not self-coincident over a finite length.

A trail is a path which may cross itself but does not reuse an edge. This sequence counts undirected trails on the square lattice up to rotation and reflection. Directed trails are counted by A006817.

Much less is known about the three-dimensional problem.

Examples

			._. ._._._. Here are the
|_. . ._. . 4 solutions
._._| . |_. when n=3 (described by 00, RR, 0L, RL).
The 24 solutions for n=5 are 0000, 000R, 00R0, 00RR, 00RL, L00L, L00R, 0R0R, 0R0L, 0RR0, 0RL0, 0LRL, 0LRR, 0LLR, 0LLL, R0LR, R0LL, R0RL, R0RR, LRLR, LRLL, LRLR, LRRR, LLRR.
		

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

• R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51.
• R. M. Foster, Solution to Problem E185, Amer. Math. Monthly, 44 (1937), 50-51. [Annotated scanned copy]
• Jessica Gonzalez, Illustration of a(4)=10
• Eric Weisstein's World of Mathematics, Self-avoiding walk.
• Index entries for sequences obtained by enumerating foldings

Crossrefs

The total number of different shapes (including those shapes where the wire is self-coincident over a finite path) is given by A001998.

Cf. A006817.

Extensions

More terms from David W. Wilson, Jul 18 2001