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.

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, 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, 6, 6, 7, 6, 6, 7, 6, 7, 8, 7, 6, 7, 7, 6, 7, 7, 5, 7

Offset: 1

Author

Clark Kimberling, May 02 2015

Keywords

• nonn
• tabl
• easy

Comments

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).

Examples

			First ten rows:
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  5  5
4  5  4  6  7  6  5  6  5
5  5  5  5  7  7  5  6  6  6
Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths are as shown here:
(2,0) -> (0,0) (1 step)
(1,1) -> (0,1) -> (1,0) -> (0,0) (3 steps)
(0,2) -> (2,0) -> (0,0) (2 steps)
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A078008, A257569, A253718.

Programs

Mathematica

f[{x_, y_}] := If[IntegerQ[x/3], {y, x/3}, {x - Mod[x, 3], y}];
g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
h[{x_, y_}] := -1 + Length[g[{x, y}]];
t = Table[h[{n - k, k}], {n, 0, 20}, {k, 0, n}];
TableForm[t] (* A253281 array *)
Flatten[t]   (* A253281 sequence *)