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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 middle of the box's edge.
4

%I #34 Feb 21 2021 02:09:51

%S 4,12,4,40,14,4,118,54,14,4,358,208,56,14,4,936,826,224,56,14,4,2600,

%T 3232,936,226,56,14,4,6212,12688,3862,956,226,56,14,4,16068,48924,

%U 16196,4026,958,226,56,14,4,34936,187276,67346,17246,4050,958,226,56,14,4

%N 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 middle of the box's edge.

%F For n <= h, T(h,n) = A259808(n).

%F Row 1 = T(1,n) = A335806(n).

%F 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.

%e T(1,2) = 12. A first step along either edge leading to the corner leaves two possible second steps. A first step to the center of either face can be followed by a second step to three edges or to the center of the box, four steps in all. Thus the total number of 2-step walks is 2*2+2*4 = 12.

%e .

%e The table begins:

%e .

%e 4 12 40 118 358 936 2600 6212 16068 34936 83708 163452 357056...

%e 4 14 54 208 826 3232 12688 48924 187276 705196 2627950 9670620 35231628...

%e 4 14 56 224 936 3862 16196 67346 282676 1180326 4950936 20646098 86165926...

%e 4 14 56 226 956 4026 17246 73588 316456 1358518 5860464 25266192 109288486...

%e 4 14 56 226 958 4050 17478 75288 327778 1425340 6236152 27260378 119641050...

%e 4 14 56 226 958 4052 17506 75600 330362 1444544 6360718 28020896 123963354...

%e 4 14 56 226 958 4052 17508 75632 330766 1448280 6391426 28238732 125405300...

%e 4 14 56 226 958 4052 17508 75634 330802 1448788 6396618 28285548 125766436...

%e 4 14 56 226 958 4052 17508 75634 330804 1448828 6397242 28292536 125835068...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293288 125844228...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845120...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845172...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293338 125845174...

%Y Cf. A259808 (h->infinity), A335806 (h=1), A337023 (start at center of box), A337031 (start at center of face), A337035 (start at corner of box), A001412, A039648.

%K nonn,walk,tabl

%O 1,1

%A _Scott R. Shannon_, Aug 14 2020