A347990 Number of n-step self-avoiding walks on a 2D square lattice where the walk cannot step to the smaller square ring of numbers than the ring it is currently on.

1, 4, 12, 36, 92, 252, 628, 1644, 4052, 10340, 25332, 63708, 155452, 387036, 941948, 2328740, 5657236, 13914596, 33757804, 82713164, 200467108, 489746916, 1186060492, 2891000036, 6997192716, 17025058164, 41186981772, 100070851212, 242000513660, 587312389940

Offset: 0

Author

Scott R. Shannon, Sep 23 2021

Keywords

• nonn
• walk

Comments

The number of the square ring around the origin the walk is currently on is just the maximum of the absolute values of its current x and y coordinates. In this sequence the SAW cannot step to a coordinate that has a smaller ring number than the ring it is currently on. For example, a step from (1,2) to either (2,2), (1,3), (0,2) is permitted as it stays on the second ring or steps to the third, but a step from (1,2) to (1,1) is forbidden as that would be stepping to the smaller first ring.

Examples

			a(0..3) are the same as the standard square lattice SAW of A001411 as the walk cannot step to a smaller ring in the first three steps.
a(4) = 92. If we restrict the first one or more steps to the right followed by an upward step then there is one walk which steps to a smaller ring and is thus forbidden. That is the walk (0,0) -> (1,0) -> (2,0) -> (2,1) -> (1,1). As this can be walked in eight different ways on the square lattice the number of 4-step walks becomes A001411(4) - 8 = 100 - 8 = 92.
		

Crossrefs

Cf. A001411, A001333, A337353.