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Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section w x w where the walk starts at the tube's edge
2

%I #15 Feb 01 2021 00:16:43

%S 4,12,4,36,14,4,98,54,14,4,274,200,56,14,4,702,744,224,56,14,4,1854,

%T 2626,926,226,56,14,4,4614,9186,3738,956,226,56,14,4,11778,31122,

%U 15056,4014,958,226,56,14,4,28914,105766,59092,17074,4050,958,226,56,14,4

%N Table read by antidiagonals: T(w,n) is the number of n-step self avoiding walks on a 3D cubic lattice confined inside a tube of cross section w x w where the walk starts at the tube's edge

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

%e T(1,2) = 12 as after a step in one of the two directions toward the adjacent tube edge the walk must turn along the side; this eliminates the 2-step straight walk in those two directions, so the total number of walks is 2*3 + 2*4 - 2 = 12.

%e The table begins:

%e 4 12 36 98 274 702 1854 4614 11778 28914 72394 176310 435346...

%e 4 14 54 200 744 2626 9186 31122 105766 351798 1175726 3859350 12729142...

%e 4 14 56 224 926 3738 15056 59092 230254 881850 3367124 12712194 47952018...

%e 4 14 56 226 956 4014 17074 71774 301578 1251362 5170636 21143094 86148002...

%e 4 14 56 226 958 4050 17464 75060 325064 1400650 6040372 25882446 110668184...

%e 4 14 56 226 958 4052 17506 75584 330070 1440668 6321926 27685144 121407404...

%e 4 14 56 226 958 4052 17508 75632 330748 1447916 6386092 28180426 124857572...

%e 4 14 56 226 958 4052 17508 75634 330802 1448768 6396174 28278426 125681952...

%e 4 14 56 226 958 4052 17508 75634 330804 1448828 6397220 28292004 125825794...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397286 28293264 125843600...

%e 4 14 56 226 958 4052 17508 75634 330804 1448830 6397288 28293336 125845094...

%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. A337400 (start at middle of tube), A337401 (start at center of tube's side) A259808 (w->infinity), A001412, A337023, A259808, A039648.

%K nonn,walk,tabl

%O 1,1

%A _Scott R. Shannon_, Aug 26 2020