This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A066372 #39 Nov 04 2024 06:05:24 %S A066372 1,1,2,3,5,8,15,23,43,71,128,209,379,650,1145,1928,3422,5908,10295, %T A066372 17530,30738,53088,91971,157194,273621,471865,814557,1393822,2414895, %U A066372 4157492,7160018,12253782,21163410,36381025,62549316,107029982,184430758 %N A066372 Number of different shapes formed by bending a piece of wire of length n in the plane. %C A066372 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. %C A066372 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 %D A066372 Deborah Freedman, dlf(AT)alumni.princeton.edu, personal communication. %H A066372 Erich Friedman, <a href="/A066372/a066372.gif">Illustration of initial terms</a> %H A066372 Ron Knott, <a href="https://r-knott.surrey.ac.uk/Fibonacci/fibforgery.html#rightlinks">Watch Out for Fibonacci Forgeries - Right-Angled Links?</a> %H A066372 <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a> %e A066372 Let LRUD denote left, right, up, down. Then for n = 1..4 the solutions are R, RD, RDL, RDR, RDLU, RDLD, RDRD. %e A066372 For n=5 the 5 shapes are: %e A066372 __.__. __. .__ |__. .__. __. %e A066372 |__| |__| |__| | |__| |__. %e A066372 |__ %Y A066372 See A001997 for another version. %Y A066372 Cf. A046661, A122224 for self-avoiding paths. %K A066372 more,nonn,nice %O A066372 1,3 %A A066372 Richard D. Plotz (Dick(AT)Plotz.com), Dec 22 2001 %E A066372 a(10)-a(23) from _Nathaniel Johnston_, Jan 04 2011 %E A066372 a(24)-a(37) from _Bert Dobbelaere_, Jan 12 2020