A374300 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 4 with a trapped endpoint.

5, 44, 330, 2231, 14234, 87670, 526549, 3105097, 18061476, 103955447, 593388315, 3364743202, 18977238539, 106562551704, 596209056866, 3325672377580, 18503794814297, 102734584002260, 569364274759972, 3150649232873918, 17411856639412771, 96118767225465184

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}. 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..22.

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.: (-(11*x^12+4*x^11-138*x^10+205*x^9+119*x^8-552*x^7+485*x^6-93*x^5-112*x^4+132*x^3-85*x^2+31*x-5)*x)/((x^6+2*x^5-9*x^4-5*x^3+15*x^2-8*x+1)*(2*x^5+3*x^4-7*x^3+12*x^2-7*x+1)).

Example

The a(1) = 5 walks are:
 *--*  *      *--*  *      *--*  *      *  *  *      *--*  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *

Crossrefs

Cf. A078528, A374299.

Sequence in context: A068311 A109984 A247776 * A227640 A096355 A371787

Adjacent sequences: A374297 A374298 A374299 * A374301 A374302 A374303

Keyword

nonn,easy

Author

Jay Pantone, Jul 16 2024

Status

approved