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 A334720 #28 Mar 17 2025 21:10:02 %S A334720 0,0,0,0,0,0,8,24,0,0,40,112,0,0,1376,2008,0,0,21720,60848,0,0,635544, %T A334720 1517368,0,0,20008456,46010640,0,0,640819936,1571759136,0,0, %U A334720 22704325648,55436103264 %N A334720 Number of 2-dimensional closed-loop self-avoiding paths on a square lattice where each path consists of steps with incrementing length from 1 to n. %C A334720 This sequence gives the number of closed-loop self avoiding walks on a 2D square lattice where the walk starts with a step length of 1 which then increments by 1 after each step up until the step length is n. No closed-loop path is possible until n = 7. %C A334720 Like A010566 all possible paths are counted, including those that are equivalent via rotation and reflection. %C A334720 For n = 8, 15, 20, 24, 27, 32, 35, 39, 44, ... = A380867, the path can be a rectangle. The first two cases are illustrated through the "Images" link from _Scott R. Shannon_. These numbers correspond to triangular numbers T(n) for which there are n1 > n2 > n3 > n4 >= 0 such that T(n) = 2(A+B) for A = T(n1) - T(n2) = T(n3) - T(n4) and B = T(n2) - T(n3). See A380867 for more. - _M. F. Hasler_, Mar 14 2025 %H A334720 A. J. Guttmann and I. G. Enting, <a href="https://doi.org/10.1088/0305-4470/21/3/009">The size and number of rings on the square lattice</a>, J. Phys. A 21 (1988), L165-L172. %H A334720 Scott R. Shannon, <a href="/A334720/a334720.txt">Images of the closed-loops for n=7,8,11,12,15</a>. %e A334720 a(1) to a(6) = 0 as no closed-loop is possible. %e A334720 a(7) = 8 as there is one path which forms a closed loop which can be walked in 8 different ways on a 2D square lattice. The path is: %e A334720 . %e A334720 5 %e A334720 *---.---.---.---.---* %e A334720 | | %e A334720 . . %e A334720 | | %e A334720 . . 4 %e A334720 | | %e A334720 6 . . %e A334720 | | 3 %e A334720 . *---.---.---* %e A334720 | | %e A334720 . . 2 %e A334720 | | %e A334720 *---.---.---.---.---.---.---X---* %e A334720 7 1 %e A334720 . %e A334720 See the attached link for text images of the closed loops for other n values. %Y A334720 Cf. A010566, A334877, A002931, A334756, A380867. %K A334720 nonn,more,walk %O A334720 1,7 %A A334720 _Scott R. Shannon_, May 08 2020