A374298 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 3 with a trapped endpoint.

2, 10, 40, 148, 526, 1828, 6256, 21190, 71260, 238432, 794914, 2643352, 8773684, 29082010, 96303640, 318678388, 1053993646, 3484654468, 11517602176, 38060746390, 125756057260, 415464635392, 1372477613794, 4533688494712, 14975452784164, 49464657237610

Offset

1,1

Comments

A growing self-avoiding walk (GSAW) is a walk on a graph that is directed, does not visit the same vertex twice, and for which all neighbors of the endpoint are part of the walk, i.e., the endpoint is trapped. This sequence is about GSAWs on the grid graph of integer points (x,y) where x >= 0 and y is in {0,1,2}. The GSAW must start at the point (0,0). The displacement of a GSAW is the difference between the largest and smallext x-values that it reaches.

Links

Table of n, a(n) for n=1..26.

Jay Pantone, A. R. Klotz, and E. Sullivan, Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height., arXiv:2407.18205 [math.CO], 2024.

Formula

G.f.: (-2*x*(x+1)*(x^3+x-1))/((x^2+2*x-1)*(x^2+3*x-1)).

Example

The a(1) = 2 walks are:
  *--*  *      *--*  *
     |         |  |
  *--*  *      *  *  *
  |               |
  *  *  *      *--*  *

Crossrefs

Cf. A078528, A374297.

Sequence in context: A174395 A320526 A193519 * A268329 A223095 A052978

Adjacent sequences: A374295 A374296 A374297 * A374299 A374300 A374301

Keyword

nonn

Author

Jay Pantone, Jul 03 2024

Status

approved