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A336818
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 size 2b X 2b where the walk starts at the middle of the box.
2
4, 8, 4, 8, 12, 4, 8, 32, 12, 4, 8, 64, 36, 12, 4, 8, 104, 96, 36, 12, 4, 8, 176, 240, 100, 36, 12, 4, 8, 296, 520, 280, 100, 36, 12, 4, 0, 496, 1048, 728, 284, 100, 36, 12, 4, 0, 848, 2104, 1816, 776, 184, 100, 36, 12, 4, 0, 1392, 4168, 4176, 2112, 780, 284, 100, 36, 12, 4
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) = A001411(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) = 8. The one 3-step walk taking a first step to the right followed by a step upward is:
.
+--+
|
*--+
.
This walk can take a downward second step, and also have a first step in the four possible directions, given a total of 1*2*4 = 8 total walks.
.
The table begins:
.
4 8 8 8 8 8 8 8 0 0 0 0 0 0 0...
4 12 32 64 104 176 296 496 848 1392 2280 3624 5472 8200 10920...
4 12 36 96 240 520 1048 2104 4168 8288 16488 32536 64680 126560 248328...
4 12 36 100 280 728 1816 4176 9304 20400 44216 95680 206104 442984 953720...
4 12 36 100 284 776 2112 5448 13704 32824 77232 178552 409144 932152 2113736...
4 12 36 100 284 780 2168 5848 15672 40472 102816 252992 615328 1472808 3501200...
4 12 36 100 284 780 2172 5912 16192 43360 115328 298856 765864 1919328 4770784...
4 12 36 100 284 780 2172 5916 16264 44016 119392 318328 843848 2194920 5664648...
4 12 36 100 284 780 2172 5916 16268 44096 120200 323856 872920 2321600 6146400...
4 12 36 100 284 780 2172 5916 16268 44100 120288 324832 880232 2363520 6344240...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324928 881392 2372968 6402928...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881496 2374328 6414896...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374440 6416472...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416592...
4 12 36 100 284 780 2172 5916 16268 44100 120292 324932 881500 2374444 6416596...
...
CROSSREFS
Cf. A001411 (b->infinity), A336872 (start on edge of box), A116903, A038373.
Sequence in context: A105398 A005883 A055026 * A205681 A059163 A091198
KEYWORD
nonn,walk,tabl
AUTHOR
Scott R. Shannon, Aug 06 2020
STATUS
approved