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A336872
Table read by antidiagonals: T(b,n) is the number of n-step self avoiding walks on a 2D square grid confined inside a square box of dimension 2b X 2b where the walk starts at the middle of one of the box's edges.
1
3, 5, 3, 10, 7, 3, 10, 17, 7, 3, 16, 39, 19, 7, 3, 10, 84, 47, 19, 7, 3, 14, 174, 119, 49, 19, 7, 3, 0, 336, 273, 129, 49, 19, 7, 3, 0, 634, 656, 325, 131, 49, 19, 7, 3, 0, 1072, 1500, 809, 337, 131, 49, 19, 7, 3, 0, 1856, 3496, 1979, 883, 339, 131, 49, 19, 7, 3
OFFSET
1,1
LINKS
A. R. Conway et al., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys A 26 (1993) 1519-1534.
A. J. Guttmann and A. R. Conway, Self-Avoiding Walks and Polygons, Annals of Combinatorics 5 (2001) 319-345.
FORMULA
For n <= b, T(b,n) = A116903(n).
For n >= b^2, T(b,n) = 0 as the walks have more steps than there are free grid points inside the box.
EXAMPLE
T(1,3) = 10. The five 3-step walks taking a first step to the right or upward steps followed by a step to the right are:
.
+ + +--+
| | |
+ +--+ +--+ +--+ +
| | | | | |
*--+ *--+ * + * *
.
This walk can also take similar steps to the left, given a total of 5*2 = 10 walks.
.
The table begins:
.
3 5 10 10 16 10 14 0 0 0 0 0 0 0 0 0...
3 7 17 39 84 174 336 634 1072 1856 2888 4598 6526 9198 11504 13758...
3 7 19 47 119 273 656 1500 3496 7612 16762 34214 71932 140664 286522 540490...
3 7 19 49 129 325 809 1979 4816 11682 28250 67606 159380 370530 842432 1902126...
3 7 19 49 131 337 883 2227 5669 14017 35108 86440 215214 528312 1303650 3162374...
3 7 19 49 131 339 897 2327 6049 15485 39421 99651 251064 631044 1583740 3969304...
3 7 19 49 131 339 899 2343 6179 16039 41809 107261 276041 701555 1790848 4530538...
3 7 19 49 131 339 899 2345 6197 16203 42585 110963 288833 746717 1925057 4942513...
3 7 19 49 131 339 899 2345 6199 16223 42787 112015 294345 767319 2003283 5188119...
3 7 19 49 131 339 899 2345 6199 16225 42809 112259 295733 775251 2035247 5318433...
3 7 19 49 131 339 899 2345 6199 16225 42811 112283 296023 777041 2046335 5366435...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296049 777381 2048599 5381553...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777409 2048993 5384369...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049023 5384821...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384853...
3 7 19 49 131 339 899 2345 6199 16225 42811 112285 296051 777411 2049025 5384855...
...
CROSSREFS
Cf. A116903 (b->infinity), A336818 (start at middle of box), A001411, A038373.
Sequence in context: A076842 A077862 A134061 * A167221 A177930 A242034
KEYWORD
nonn,walk,tabl
AUTHOR
Scott R. Shannon, Aug 06 2020
STATUS
approved