A374302 Number of growing self-avoiding walks with displacement n on a half-infinite strip of height 5 with a trapped endpoint.

11, 172, 2329, 28130, 318086, 3454914, 36484161, 377467377, 3845503176, 38709658128, 385953901159, 3818368690421, 37534770596896, 366993128166171, 3571984859121359, 34631980574240256, 334654089341585090, 3224481296529386602, 30990605791226254096

Offset: 1

Author

Jay Pantone, Jul 22 2024

Keywords

• nonn

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}. 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.

Examples

			Five of the a(1) = 11 walks are:
 *--*  *      *--*  *      *--*  *      *  *  *      *--*  *
 |  |         |  |         |  |                      |  |
 *  *  *      *  *  *      *  *  *      *  *  *      *  *  *
    |         |  |            |                      |  |
 *--*  *      *  *  *      *--*  *      *--*  *      *  *  *
 |               |         |            |  |         |  |
 *  *  *      *--*  *      *--*  *      *  *  *      *  *  *
 |            |               |            |            |
 *  *  *      *  *  *      *--*  *      *--*  *      *--*  *
		

Links

• 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.

Crossrefs

Cf. A078528, A374301.

Formula

G.f.: (x*(970*x^26 + 16189*x^25 + 76977*x^24 + 107296*x^23 - 167471*x^22 - 441374*x^21 + 302374*x^20 + 825566*x^19 - 591839*x^18 - 531077*x^17 + 861370*x^16 - 734832*x^15 - 170227*x^14 + 1369959*x^13 - 918040*x^12 - 622581*x^11 + 986287*x^10 - 181528*x^9 - 333951*x^8 + 247985*x^7 - 57814*x^6 - 11881*x^5 + 13594*x^4 - 5279*x^3 + 1221*x^2 - 169*x + 11))/((3*x^14 + 23*x^13 + 74*x^12 + 130*x^11 - 118*x^10 - 96*x^9 - 260*x^8 + 362*x^7 + 500*x^6 - 650*x^5 - 27*x^4 + 237*x^3 - 105*x^2 + 18*x - 1)*(28*x^11 + 50*x^10 - 48*x^9 - 112*x^8 + 140*x^7 + 151*x^6 - 209*x^5 - 17*x^4 + 66*x^3 - 45*x^2 + 13*x - 1)).