A257570 - OEIS

%I #5 May 02 2015 10:15:55

%S 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 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 3,3,4,4,4,4,3,3,4,6,5,5,3,4,1,5,0,5

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

%C 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 Clark Kimberling, <a href="/A257570/b257570.txt">Antidiagonals n = 1..60, flattened</a>

%e Northwest corner:

%e 0  1  2  3  3  4  4  5  4  5  5

%e 1  0  1  2  2  3  3  4  3  4  4

%e 2  1  0  1  1  2  2  3  2  3  3

%e 3  2  1  0  2  3  1  2  3  4  4

%e 3  2  1  2  0  1  3  4  1  2  2

%e 4  3  2  3  1  0  4  5  2  3  1

%e d(4,6) = d(6,4) = 3 counts the edges in the path 6,3,2,4;

%e d(46,21) = 6 counts the edges in the path 46,23,22,11,10,20,21.

%t f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];

%t g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];

%t s[n_] := Reverse[Select[Sort[Flatten[Select[g[{n, 0}], #[[2]] == 0 &]]], # > 0 &]];

%t m[h_, k_] := Max[Intersection[s[h], s[k]]];

%t j[h_, k_] := Join[Select[s[h], # >= m[h, k] &], Reverse[Select[s[k], # > m[h, k] &]]];

%t d[h_, k_] := If[k*h == 0, Length[j[h, k]], -1 + Length[j[h, k]]];

%t TableForm[Table[d[h, k], {h, 0, 16}, {k, 0, 16}]]  (* A257570 array *)

%t Flatten[Table[d[h - k, k], {h, 0, 20}, {k, 0, h}]  (* A257570 sequence *)]

%Y Cf. A257571, A232558.

%K nonn,tabl,easy

%O 1,4

%A _Clark Kimberling_, May 01 2015