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A374297
Number of growing self-avoiding walks of length n on a half-infinite strip of height 3 with a trapped endpoint.
2
1, 2, 2, 6, 10, 20, 41, 79, 146, 285, 538, 1039, 1982, 3812, 7272, 13961, 26686, 51161, 97865, 187518, 358835, 687327, 1315616, 2519472, 4823116, 9235610, 17681264, 33855310, 64817361, 124105590, 237610012, 454943624, 871035486, 1667726103, 3193049603
OFFSET
4,2
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}. 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.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,0,-1,2,-4,-3,-2,0,-4,-4).
FORMULA
G.f.: x^4*(1 + x - 2*x^2 - x^5 + x^6 - 2*x^8 - 5*x^9 - 5*x^10 - 2*x^11 - 2*x^12)/((1 + x^4)*(1 - 2*x^2)*(1 - x - 2*x^3 - x^4 - 2*x^5 - 2*x^6)).
EXAMPLE
The a(4) = 1 and a(5) = 2 walks are:
*--* * *--* * * * *
| | |
*--* * * * * *--*--*
| | | |
* * *--* * * *--*
The GSAW below has length 10.
*--*--* * * *
|
*--* *--* * *
| | |
* *--*--* * *
CROSSREFS
Cf. A078528.
Sequence in context: A123757 A167399 A247326 * A375188 A300274 A019310
KEYWORD
nonn,easy
AUTHOR
Jay Pantone, Jul 03 2024
STATUS
approved