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A347990
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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.
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4
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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