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.
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
Examples
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... ...
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.