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A253281 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.

%I A253281 #9 May 06 2015 09:48:05
%S A253281 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 A253281 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 A253281 6,6,7,6,6,7,6,7,8,7,6,7,7,6,7,7,5,7
%N A253281 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 A253281 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 A253281 Clark Kimberling, <a href="/A253281/b253281.txt">Table of n, a(n) for n = 1..1000</a>
%e A253281 First ten rows:
%e A253281 0
%e A253281 1  2
%e A253281 1  3  2
%e A253281 3  3  3  4
%e A253281 4  4  3  5  5
%e A253281 4  5  4  5  6  5
%e A253281 3  5  5  5  6  6  4
%e A253281 4  4  5  6  6  6  5  5
%e A253281 4  5  4  6  7  6  5  6  5
%e A253281 5  5  5  5  7  7  5  6  6  6
%e A253281 Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths are as shown here:
%e A253281 (2,0) -> (0,0) (1 step)
%e A253281 (1,1) -> (0,1) -> (1,0) -> (0,0) (3 steps)
%e A253281 (0,2) -> (2,0) -> (0,0) (2 steps)
%t A253281 f[{x_, y_}] := If[IntegerQ[x/3], {y, x/3}, {x - Mod[x, 3], y}];
%t A253281 g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
%t A253281 h[{x_, y_}] := -1 + Length[g[{x, y}]];
%t A253281 t = Table[h[{n - k, k}], {n, 0, 20}, {k, 0, n}];
%t A253281 TableForm[t] (* A253281 array *)
%t A253281 Flatten[t]   (* A253281 sequence *)
%Y A253281 Cf. A078008, A257569, A253718.
%K A253281 nonn,tabl,easy
%O A253281 1,3
%A A253281 _Clark Kimberling_, May 02 2015