%I #9 May 06 2015 09:48:05

%S 0,1,2,1,3,2,3,3,3,4,4,4,3,5,5,4,5,4,5,6,5,3,5,5,5,6,6,4,4,4,5,6,6,6,

%T 5,5,4,5,4,6,7,6,5,6,5,5,5,5,5,7,7,5,6,6,6,6,6,5,6,6,7,6,6,6,7,7,6,7,

%U 6,6,7,6,6,7,6,7,8,7,6,7,7,6,7,7,5,7

%N Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where allowable steps are as follows: (x,y) -> (x-r, y) if r > 0, and (x,y) -> (y, r/3) otherwise, where r = x mod 3.

%C For n >= 3, the number of pairs (h,k) satisfying T(h,k) = n is A078008(n+1) for n >= 0. The number of pairs of the form (h,0) satisfying T(h,0) = n is A253718(n).

%H Clark Kimberling, <a href="/A253281/b253281.txt">Table of n, a(n) for n = 1..1000</a>

%e First ten rows:

%e 0

%e 1 2

%e 1 3 2

%e 3 3 3 4

%e 4 4 3 5 5

%e 4 5 4 5 6 5

%e 3 5 5 5 6 6 4

%e 4 4 5 6 6 6 5 5

%e 4 5 4 6 7 6 5 6 5

%e 5 5 5 5 7 7 5 6 6 6

%e Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths are as shown here:

%e (2,0) -> (0,0) (1 step)

%e (1,1) -> (0,1) -> (1,0) -> (0,0) (3 steps)

%e (0,2) -> (2,0) -> (0,0) (2 steps)

%t f[{x_, y_}] := If[IntegerQ[x/3], {y, x/3}, {x - Mod[x, 3], y}];

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

%t h[{x_, y_}] := -1 + Length[g[{x, y}]];

%t t = Table[h[{n - k, k}], {n, 0, 20}, {k, 0, n}];

%t TableForm[t] (* A253281 array *)

%t Flatten[t] (* A253281 sequence *)

%Y Cf. A078008, A257569, A253718.

%K nonn,tabl,easy

%O 1,3

%A _Clark Kimberling_, May 02 2015