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A048248
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Possible traces of n-step walks on 1-D lattice, ignoring translations.
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1
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1, 2, 4, 7, 14, 23, 45, 72, 137, 217, 402, 635, 1149, 1816, 3221, 5101, 8898, 14127, 24305, 38688, 65809, 105009, 176962, 282995, 473269, 758312, 1260253, 2022661, 3344354, 5375207
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OFFSET
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1,2
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COMMENTS
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Translations discounted, so the sequence of visits <0,1,2,1,0> has trace (2,2), as do <0,-1,0,1,0>, <0,-1,-2,-1,0>, etc.
Trace as used here refers to the number of times an edge is used in the walk. - Sean A. Irvine, Jun 06 2021
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LINKS
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EXAMPLE
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a(4)=7 since a walk of 4 steps can leave traces (1,1,1,1), (1,1,2), (2,1,1), (2,2), (1,3), (3,1) and (4). Note that (1,2,1) is impossible.
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MATHEMATICA
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For[size = 1, size < 10, size++, traces = {}; For[i = 0, i < 2^ size, i++, thePath = IntegerDigits[i, 2, size]*2 - 1; loc = size + 2; theTrace = Table[0, {z, -size - 1, size + 1}]; For[j = 1, j <= size, j++, loc += thePath[[j]]; If[thePath[[j]] == 1, theTrace[[loc - 1]]++, theTrace[[loc]]++ ]; ]; theTrace = Select[theTrace, # > 0 &]; If[ ! MemberQ[traces, theTrace], traces = Append[traces, theTrace]]; ]; Print[Length[traces]]]; - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(19)-a(21) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006
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STATUS
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approved
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