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

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, 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, 3, 3, 4, 4, 4, 4, 3, 3, 4, 6, 5, 5, 3, 4, 1, 5, 0, 5

Offset: 1

Author

Clark Kimberling, May 01 2015

Keywords

• nonn
• tabl
• easy

Comments

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.

Examples

			Northwest corner:
0  1  2  3  3  4  4  5  4  5  5
1  0  1  2  2  3  3  4  3  4  4
2  1  0  1  1  2  2  3  2  3  3
3  2  1  0  2  3  1  2  3  4  4
3  2  1  2  0  1  3  4  1  2  2
4  3  2  3  1  0  4  5  2  3  1
d(4,6) = d(6,4) = 3 counts the edges in the path 6,3,2,4;
d(46,21) = 6 counts the edges in the path 46,23,22,11,10,20,21.
		

Links

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

Crossrefs

Cf. A257571, A232558.

Programs

Mathematica

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
s[n_] := Reverse[Select[Sort[Flatten[Select[g[{n, 0}], #[[2]] == 0 &]]], # > 0 &]];
m[h_, k_] := Max[Intersection[s[h], s[k]]];
j[h_, k_] := Join[Select[s[h], # >= m[h, k] &], Reverse[Select[s[k], # > m[h, k] &]]];
d[h_, k_] := If[k*h == 0, Length[j[h, k]], -1 + Length[j[h, k]]];
TableForm[Table[d[h, k], {h, 0, 16}, {k, 0, 16}]]  (* A257570 array *)
Flatten[Table[d[h - k, k], {h, 0, 20}, {k, 0, h}]  (* A257570 sequence *)]