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}. 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
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
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)).
EXAMPLE
Five of the a(1) = 11 walks are:
*--* * *--* * *--* * * * * *--* *
| | | | | | | |
* * * * * * * * * * * * * * *
| | | | | |
*--* * * * * *--* * *--* * * * *
| | | | | | |
* * * *--* * *--* * * * * * * *
| | | | |
* * * * * * *--* * *--* * *--* *
CROSSREFS
KEYWORD
nonn
AUTHOR
Jay Pantone, Jul 22 2024
STATUS
approved