A337033 The number of n-step self avoiding walks on a 3D cubic lattice confined inside a box of size 2x2x2 where the walk starts at the center of one of the box's faces.

1, 5, 17, 52, 148, 400, 1060, 2700, 6720, 15760, 36248, 77856, 163296, 312760, 590536, 995160, 1663664, 2405056, 3482320, 4180656, 5080320, 4823560, 4686432, 3165088, 2228584, 792272, 303264, 0

Offset: 0

Author

Scott R. Shannon, Aug 12 2020

Keywords

• nonn
• walk
• fini
• full

Examples

			a(1) = 5 as the walk is free to move one step in five possible directions. It cannot take a step to a direction opposite to the face's normal it starts on.
a(2) = 17. Taking the first along the starting face hits the box's edge after which the walks has three directions for the second step, giving 4*3 = 12 walks in all. A first step away from the starting face can be followed by a second step in five directions. The total number of 2-step walks is therefore 12+5 = 17.
a(26) = 303264. This is the total number of ways a 26-step walk can completely fill the 2x2x2 box's 26 available lattice points. Unlike the walk which starts at the center of the box, see A337021, all lattice points can be visited in one walk.
		

Crossrefs

Cf. A337031 (other box sizes), A337021 (start at center of box), A335806 (start at middle of edge), A337034 (start at corner of box), A001412, A116904.

Formula

For n>=27 all terms are 0 as the walk contains more steps than there are available lattice points in the 2x2x2 box.