cp's OEIS Frontend

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

%I A001997 M1206 N0465 #39 Jul 08 2025 16:27:20
%S A001997 1,1,2,4,10,24,66,176,493,1362,3821,10660,29864,83329,232702,648182,
%T A001997 1804901,5015725,13931755,38635673,107090666,296449133,820271143,
%U A001997 2267225157,6264244414,17291930470
%N A001997 Number of different shapes formed by bending a piece of wire of length n in the plane.
%C A001997 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.
%C A001997 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.
%C A001997 Much less is known about the three-dimensional problem.
%D A001997 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001997 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001997 R. M. Foster, <a href="http://www.jstor.org/stable/2301339">Solution to Problem E185</a>, Amer. Math. Monthly, 44 (1937), 50-51.
%H A001997 R. M. Foster, <a href="/A001997/a001997.pdf">Solution to Problem E185</a>, Amer. Math. Monthly, 44 (1937), 50-51. [Annotated scanned copy]
%H A001997 Jessica Gonzalez, <a href="/A001997/a001997.png">Illustration of a(4)=10</a>
%H A001997 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Self-AvoidingWalk.html">Self-avoiding walk.</a>
%H A001997 <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%e A001997 ._. ._._._. Here are the
%e A001997 |_. . ._. . 4 solutions
%e A001997 ._._| . |_. when n=3 (described by 00, RR, 0L, RL).
%e A001997 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.
%Y A001997 The total number of different shapes (including those shapes where the wire is self-coincident over a finite path) is given by A001998.
%Y A001997 Cf. A006817.
%K A001997 nonn,more,nice,walk
%O A001997 0,3
%A A001997 _N. J. A. Sloane_
%E A001997 More terms from _David W. Wilson_, Jul 18 2001