A066372 Number of different shapes formed by bending a piece of wire of length n in the plane.

1, 1, 2, 3, 5, 8, 15, 23, 43, 71, 128, 209, 379, 650, 1145, 1928, 3422, 5908, 10295, 17530, 30738, 53088, 91971, 157194, 273621, 471865, 814557, 1393822, 2414895, 4157492, 7160018, 12253782, 21163410, 36381025, 62549316, 107029982, 184430758

Offset: 1

Author

Richard D. Plotz (Dick(AT)Plotz.com), Dec 22 2001

Keywords

• more
• nonn
• nice

Comments

Wire is marked into n equal segments by n-1 marks, is bent at right angles at each of these points, making each segment parallel to one of two rectangular axes. (Stays in plane, bends are of +-90 degs.) May cross itself but is not self-coincident over a finite length. Two configurations which differ only in a rotation or turning over are not counted as different.

In addition to not allowing straight segments, there are two further subtle differences between the counting here and the counting in A001997. In this sequence, if the wire effectively forms a closed loop, then that shape is counted only once, whereas in A001997 the position of the ends of the wire matters. Similarly, the same consideration applies to places where the wire is self-coincident. In this sequence, we assume our eyes are not good enough to distinguish which of two (or more) ways of bending the wire achieve the same shape. These distinctions first matter for n=8 where in this sequence all three arrangements which look like a slanted 8 are equivalent. - Sean A. Irvine, Oct 10 2023

Examples

			Let LRUD denote left, right, up, down. Then for n = 1..4 the solutions are R, RD, RDL, RDR, RDLU, RDLD, RDRD.
For n=5 the 5 shapes are:
  __.__.  __.  .__  |__.    .__.     __.
    |__|    |__|       |__| |  |__|    |__.
                                          |__
		

References

• Deborah Freedman, dlf(AT)alumni.princeton.edu, personal communication.

Links

• Erich Friedman, Illustration of initial terms
• Ron Knott, Watch Out for Fibonacci Forgeries - Right-Angled Links?
• Index entries for sequences obtained by enumerating foldings

Crossrefs

See A001997 for another version.

Cf. A046661, A122224 for self-avoiding paths.

Extensions

a(10)-a(23) from Nathaniel Johnston, Jan 04 2011

a(24)-a(37) from Bert Dobbelaere, Jan 12 2020