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A257570 Rectangular array, read by antidiagonals: d(h,k) = distance between h and k in the tree at A232558, for h >=0, k >= 0.

%I A257570 #5 May 02 2015 10:15:55
%S A257570 0,1,1,2,0,2,3,1,1,3,3,2,0,2,3,4,2,1,1,2,4,4,3,1,0,1,3,4,5,3,2,2,2,2,
%T A257570 3,5,4,4,2,3,0,3,2,4,4,5,3,3,1,1,1,1,3,3,5,5,4,2,2,3,0,3,2,2,4,5,6,4,
%U A257570 3,3,4,4,4,4,3,3,4,6,5,5,3,4,1,5,0,5
%N A257570 Rectangular array, read by antidiagonals: d(h,k) = distance between h and k in the tree at A232558, for h >=0, k >= 0.
%C A257570 The distance between h and k is the length of the path from h to k in the tree defined from the root 0 by edges from x to x+1 and x to 2x if x is even, and an edge from x to x+1 if x is odd.  This is the tree defined at A232558; it is a subtree of the tree defined at A257569.
%H A257570 Clark Kimberling, <a href="/A257570/b257570.txt">Antidiagonals n = 1..60, flattened</a>
%e A257570 Northwest corner:
%e A257570 0  1  2  3  3  4  4  5  4  5  5
%e A257570 1  0  1  2  2  3  3  4  3  4  4
%e A257570 2  1  0  1  1  2  2  3  2  3  3
%e A257570 3  2  1  0  2  3  1  2  3  4  4
%e A257570 3  2  1  2  0  1  3  4  1  2  2
%e A257570 4  3  2  3  1  0  4  5  2  3  1
%e A257570 d(4,6) = d(6,4) = 3 counts the edges in the path 6,3,2,4;
%e A257570 d(46,21) = 6 counts the edges in the path 46,23,22,11,10,20,21.
%t A257570 f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
%t A257570 g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
%t A257570 s[n_] := Reverse[Select[Sort[Flatten[Select[g[{n, 0}], #[[2]] == 0 &]]], # > 0 &]];
%t A257570 m[h_, k_] := Max[Intersection[s[h], s[k]]];
%t A257570 j[h_, k_] := Join[Select[s[h], # >= m[h, k] &], Reverse[Select[s[k], # > m[h, k] &]]];
%t A257570 d[h_, k_] := If[k*h == 0, Length[j[h, k]], -1 + Length[j[h, k]]];
%t A257570 TableForm[Table[d[h, k], {h, 0, 16}, {k, 0, 16}]]  (* A257570 array *)
%t A257570 Flatten[Table[d[h - k, k], {h, 0, 20}, {k, 0, h}]  (* A257570 sequence *)]
%Y A257570 Cf. A257571, A232558.
%K A257570 nonn,tabl,easy
%O A257570 1,4
%A A257570 _Clark Kimberling_, May 01 2015