OFFSET
5,1
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 length of a GSAW is the number of edges.
LINKS
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.: ((12*x^39 + 14*x^38 - 20*x^37 - 18*x^36 - 45*x^35 - 12*x^34 + 107*x^33 - 38*x^32 + 3*x^31 - 49*x^30 - 38*x^29 + 242*x^28 - 11*x^27 - 66*x^26 - 181*x^25 - 144*x^24 + 246*x^23 + 91*x^22 + 72*x^21 - 208*x^20 - 150*x^19 + 98*x^18 + 57*x^17 + 143*x^16 - 74*x^15 + 5*x^14 - 21*x^13 + 28*x^12 - 17*x^11 - 55*x^10 - 17*x^9 + 22*x^8 + 45*x^7 + 10*x^6 - 19*x^5 - 21*x^4 + 3*x^3 + 7*x^2 + 4*x - 3)*x^5)/((2*x^19 + 2*x^18 - 7*x^17 - 6*x^16 + 5*x^15 + 8*x^14 + 7*x^13 - 17*x^12 - 8*x^11 + 3*x^10 + 10*x^9 + 3*x^8 - 8*x^7 + 2*x^6 - x^5 + 6*x^4 - 3*x^3 - 2*x + 1)*(4*x^20 - 2*x^18 - 5*x^16 + 8*x^14 - x^12 + 2*x^10 - 4*x^8 + 2*x^6 + 3*x^4 - 4*x^2 + 1)).
EXAMPLE
The a(5) = 3 walks are:
*--* * * * * * * *
|
*--* * *--* * * * *
| | |
* * * * * * *--*--*
| | | |
* * * *--* * * *--*
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jay Pantone, Jul 15 2024
STATUS
approved