A226324 Array by antidiagonals: D(m,n) = distance between m and n using the graph-metric of A226247.

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

Offset: 1

Author

Clark Kimberling, Jun 04 2013

Keywords

• nonn
• tabl
• easy

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.

Examples

			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.
		

Links

• Clark Kimberling, Antidiagonals n=1..60, flattened

Crossrefs

Cf. A226080, A226207, A226247.

Programs

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 *)