A337023 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 2h X 2h X 2h where the walk starts at the center of the box.

6, 24, 6, 72, 30, 6, 168, 144, 30, 6, 456, 624, 150, 30, 6, 1032, 2520, 720, 150, 30, 6, 2712, 9360, 3408, 726, 150, 30, 6, 5784, 34008, 15432, 3528, 726, 150, 30, 6, 14640, 120960, 68088, 16776, 3534, 726, 150, 30, 6, 29760, 430656, 289128, 79320, 16920, 3534, 726, 150, 30, 6

Offset: 1

Author

Scott R. Shannon, Aug 11 2020

Keywords

• nonn
• walk
• tabl

Examples

			T(1,2) = 24 as after a step in one of the six axial directions the walk must turn along the face of the box; this eliminates the 2-step straight walk in all directions, so the total number of walks is 6*5-6 = 24.
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The table begins:
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6 24  72 168  456  1032  2712   5784   14640   29760    71136    133344    291696..
6 30 144 624 2520  9360 34008 120960  430656 1511856  5340312  18587208  65176416..
6 30 150 720 3408 15432 68088 289128 1205976 4920528 19985928  80066136 321160728..
6 30 150 726 3528 16776 79320 366960 1677864 7516992 33312456 145379760 630249720..
6 30 150 726 3534 16920 81216 385224 1822584 8518920 39588480 181800312 829567656..
6 30 150 726 3534 16926 81384 387768 1850376 8765304 41478144 194837136 912538512..
6 30 150 726 3534 16926 81390 387960 1853664 8805504 41872944 198158520 937459176..
6 30 150 726 3534 16926 81390 387966 1853880 8809632 41928816 198761160 942984312..
6 30 150 726 3534 16926 81390 387966 1853886 8809872 41933880 198836352 943868424..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934144 198842448 943966968..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842736 943974192..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974504..
6 30 150 726 3534 16926 81390 387966 1853886 8809878 41934150 198842742 943974510..
		

Crossrefs

Cf. A001412 (h->infinity), A337021 (h=1), A337031 (start at center of face), A337035 (start as corner of box), A336862 (start at middle of edge), A116904, A039648.

Formula

For n <= h, T(h,n) = A001412(n).

Row 1 = T(1,n) = A337021(n).

For n >= (2h+1)^3, T(h,n) = 0 as the walk contains more steps than there are available lattice points in the 2h X 2h X 2h box.