A374306 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 7 with a trapped endpoint.

47, 2221, 94006, 3527224, 123159829, 4110628551, 133093672039, 4216993511767, 131454310596858, 4046054885054361, 123275425298494683, 3724935782123793466, 111781579014020685006, 3335061533295212856274, 99013139230297294579692, 2927094675162133314593603

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 {0,1,2,3,4,5,6}. 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..16.

Jay Pantone, Generating function.

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

Formula

See Links section for generating function.

Example

Five of the a(1) = 47 walks are:
 *--*  *      *  *  *      *  *  *      *  *  *      *--*  *
 |  |                                                |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
 |  |                                                |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |  |         |  |         |  |                      |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *

Crossrefs

Cf. A078528, A374305.

Sequence in context: A218749 A049668 A009991 * A052463 A327770 A005148

Adjacent sequences: A374303 A374304 A374305 * A374307 A374308 A374309

Keyword

nonn

Author

Jay Pantone, Jul 23 2024

Status

approved