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A336492
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Total number of neighbor contacts for n-step self-avoiding walks on a 2D square lattice.
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2
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0, 0, 8, 32, 152, 512, 1880, 5920, 19464, 59168, 183776, 545392, 1638400, 4778000, 14043224, 40422544, 116977176, 333346928, 953538440, 2695689520, 7642091352, 21464794032, 60417010152, 168787016352, 472315518008, 1313548558528, 3657850909680, 10133559518800
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OFFSET
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1,3
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COMMENTS
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This sequence gives the total number of neighbor contacts for all n-step self avoiding walks on a 2D square lattices. A neighbor contact is when the walk comes within 1 unit distance of a previously visited point, excluding the previous adjacent point.
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LINKS
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EXAMPLE
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a(1) = a(2) = 0 as a 1 and 2 step walk cannot approach a previous step.
a(3) = 8. The single walk where one interaction occurs, which can be taken in eight ways on a 2D square lattice, is:
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+---+
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X---+
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Therefore, the total number of interactions is 1*1*8 = 8.
a(4) = 32. The four walks where one interaction occurs, each of which can be taken in eight ways on a 2D square lattice, are:
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+---+---+ + +---+ +---+
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X---+ +---+ X---+---+ X---+ +
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X---+
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Therefore, the total number of interactions is 4*1*8 = 32.
a(5) = 152. Considering only walks which start with one or more steps to the right followed by an upward step there are thirty-five different walks. Eleven of these have one neighbor contact (hence A033155(5) = 11*8 = 88) while four have two contacts. These are:
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+---+---+ +---+---+ +---+ +---+
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+ X---+ X---+---+ +---+ + +
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X---+ X---+
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Therefore, the total number of contacts is (11*1 + 4*2)*8 = 152.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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