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A337035
Table read by antidiagonals: T(h,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size h x h x h where the walk starts at one of the box's corners.
4
3, 6, 3, 12, 9, 3, 18, 30, 9, 3, 30, 96, 33, 9, 3, 24, 294, 120, 33, 9, 3, 18, 840, 456, 123, 33, 9, 3, 0, 2214, 1662, 486, 123, 33, 9, 3, 0, 5796, 6018, 1908, 489, 123, 33, 9, 3, 0, 14112, 20784, 7584, 1944, 489, 123, 33, 9, 3, 0, 34158, 70470, 29754, 7932, 1947, 489, 123, 33, 9, 3
OFFSET
1,1
FORMULA
For n <= h, T(h,n) = A039648(n).
Row 2 = T(2,n) = A337034(n).
For n >= (h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the hxhxh box.
EXAMPLE
T(2,3) = 30. After the first step along the cube's edge the walk can turn toward a face center in two ways. From the face center is has four available directions. If instead the walk takes two steps along the cube's edge to another corner it then has only two directions available for a third step. As the first step can be taken in three ways the total number of 3-step walks is 3*2*4+3*2 = 30.
.
The table begins:
.
3 6 12 18 30 24 18 0 0 0 0 0 0 0...
3 9 30 96 294 840 2214 5796 14112 34158 76062 167928 337476 670626...
3 9 33 120 456 1662 6018 20784 70470 231648 754386 2396832 7562730 23297826...
3 9 33 123 486 1908 7584 29754 115866 444096 1678560 6260082 23037330 84061494...
3 9 33 123 489 1944 7932 32298 132720 541908 2212542 8946288 36007908 143452686...
3 9 33 123 489 1947 7974 32766 136590 570570 2397384 10062258 42243138 176723826...
3 9 33 123 489 1947 7977 32814 137196 576168 2443284 10386522 44376156 189622260...
3 9 33 123 489 1947 7977 32817 137250 576930 2451066 10456566 44914830 193454916...
3 9 33 123 489 1947 7977 32817 137253 576990 2452002 10467042 45017580 194310204...
3 9 33 123 489 1947 7977 32817 137253 576993 2452068 10468170 45031314 194456058...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468242 45032652 194473668...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032730 194475234...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032733 194475318...
3 9 33 123 489 1947 7977 32817 137253 576993 2452071 10468245 45032733 194475321...
CROSSREFS
Cf. A039648 (h->infinity), A337034 (h=2), A337031 (start at center of face), A337032 (start as center of box), A336862 (start at middle of edge), A001412.
Sequence in context: A160899 A360981 A336769 * A203491 A085709 A120910
KEYWORD
nonn,walk,tabl
AUTHOR
Scott R. Shannon, Aug 12 2020
STATUS
approved