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 A226324 Array by antidiagonals:  D(m,n) = distance between m and n using the graph-metric of A226247. 1
 0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 1, 3, 3, 2, 2, 2, 2, 3, 4, 2, 1, 0, 1, 2, 4, 4, 3, 1, 3, 3, 1, 3, 4, 4, 3, 2, 3, 0, 3, 2, 3, 4, 5, 3, 2, 4, 2, 2, 4, 2, 3, 5, 5, 4, 2, 4, 1, 0, 1, 4, 2, 4, 5, 5, 4, 3, 4, 1, 3, 3, 1, 4, 3, 4, 5, 5, 4, 3, 5, 3, 3, 0, 3 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Let S be the set of numbers defined by these rules:  0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x is in S.  Then S is the set of all rational numbers, produced in generations as follows: g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4,-1/3,1/2),... For n > 2, once g(n-1) = (c(1),...,c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2),...,c(z)+1, -1/c(z)) by deleting previously generated elements.  This order of generation matches a tree with (0,1), (1,2), (1,-1), (2,3), (2,-1/2), (3,4), (4,-1/3), (-1/2,1/2), etc.  Replace each node by the order in which it is generated, so that the nodes labeled (0,1,2,-1,3,-1/2,4,-1/3,...) get new labels (1,2,3,4,5,6,...), respectively.  If m and n are positive integers, then D(m,n) is the number of edges between m and n. LINKS Clark Kimberling, Antidiagonals n=1..60, flattened EXAMPLE Northwest corner of the distance table: 0 1 2 2 3 3 4 4 4 5 1 0 1 1 2 2 3 3 3 4 2 1 0 2 1 1 2 2 2 3 2 1 2 0 3 3 4 4 4 5 3 2 1 3 0 2 1 1 3 2 3 2 1 3 2 0 3 3 1 4 4 3 2 4 1 3 0 2 4 1 4 3 2 4 1 3 2 0 4 3 4 3 2 4 3 1 4 4 0 5 5 4 3 5 2 4 1 3 5 0 Row 5, column 4 is occupied by 3, meaning that D(5,4) = 3, a count of edges in the subgraph 5 -> 3 -> 2 -> 4. MATHEMATICA \$MaxExtraPrecision = Infinity; g[1] := {1}; g[2] := {1, 0}; g[3] := {1, 0, 0}; g[test_] := Module[{topRow, len, tmp = test, noOfTerms = Ceiling[Log[test]/Log[1.465571231876768026656731]] - 1}, topRow = Flatten[{1, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, noOfTerms]}]; If[First[#] == 0, Rest[#], #] &[Table[If[# >= 0, tmp = #; 1, 0] &[tmp - topRow[[n]]], {n, noOfTerms, 1, -1}]]]; d[n1_, n2_] := Module[{z1 = g[n1], z2 = g[n2]}, Length[z1] + Length[z2] - 2(NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1, Min[{Length[z1], Length[z2]}]] - 1)]; (dArray = Table[d[m, n], {m, 1, #}, {n, 1, #}] &[15]) // TableForm   Flatten[Table[d[k, n + 1 - k], {n, 1, 15}, {k, 1, n}]]   ArrayPlot[dArray, ColorFunction -> "BlueGreenYellow"] (* Peter J. C. Moses, Jun 02 2013 *) CROSSREFS Cf. A226080, A226207, A226247. Sequence in context: A129678 A261773 A226207 * A023604 A219660 A060964 Adjacent sequences:  A226321 A226322 A226323 * A226325 A226326 A226327 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Jun 04 2013 STATUS approved

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Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)