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A304584
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A linear mapping a(n) = x + d*n of pairs of nonnegative integers (x,d), where the pairs are enumerated by antidiagonals.
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5
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0, 1, 2, 2, 5, 10, 3, 9, 17, 27, 4, 14, 26, 40, 56, 5, 20, 37, 56, 77, 100, 6, 27, 50, 75, 102, 131, 162, 7, 35, 65, 97, 131, 167, 205, 245, 8, 44, 82, 122, 164, 208, 254, 302, 352, 9, 54, 101, 150, 201, 254, 309, 366, 425, 486, 10, 65, 122, 181, 242, 305, 370, 437, 506, 577, 650, 11
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OFFSET
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0,3
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COMMENTS
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The sequence solves the following riddle, which has been communicated by Klaus Nagel: A flea starts to jump on the nonnegative integers at time = 0 at an unknown location x >= 0 making jumps of unknown, but constant distance d >= 0 at every subsequent time step. By which strategy can the flea be captured with 100% certainty in a finite number of trials? The solution is to hit a(n) at time = n. This works for all enumerations of pairs (x,d) of integers, because eventually any combination of starting location x and jump width d will be addressed.
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LINKS
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EXAMPLE
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d:
5 | 20
4 | 14 19
3 | 9 13 18
2 | 5 8 12 17
1 | 2 4 7 11 16
0 | 0 1 3 6 10 15
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x: 0 1 2 3 4 5
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a(13) = 1 + 13*3 = 40 because the 13th position in the enumeration corresponds to x=1 and d=3.
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MAPLE
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pos2pair:=proc(n) local w, k, e; w:=floor(sqrt(2*n)); if w*(w+1)>2*n then k:=w-1; else k:=w; fi; e:=n-k*(k+1)/2; return [k-e, e]; end:WhereFlea:=proc(n) local x, d, pair; pair:=pos2pair(n); x:=pair[1]; d:=pair[2]; return x+d*n; end:
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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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