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

1,3

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).

Links

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

Example

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)

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 *)

Crossrefs

Cf. A078008, A257569, A253718.

Sequence in context: A263100 A261388 A369793 * A029206 A029200 A317243

Adjacent sequences: A253278 A253279 A253280 * A253282 A253283 A253284

Keyword

nonn,tabl,easy

Author

Clark Kimberling, May 02 2015

Status

approved