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.

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

Links

Table of n, a(n) for n=1..66.

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.
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The table begins:
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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

Adjacent sequences: A337032 A337033 A337034 * A337036 A337037 A337038

Keyword

nonn,walk,tabl

Author

Scott R. Shannon, Aug 12 2020

Status

approved