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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.

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proposed

Triangular array: T(h,k) = number of steps from (h,k) to (0,0), where allowable steps are as follows: (x,y) -> (x-(x mod 3), r, y) if (x mod 3) is not r > 0, and (x,y) -> (y, xr/3) otherwise, where r = x mod 3.

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).

The number of pairs of the form (h,0) satisfying T(h,0) = n is A253718(n).

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}];

Flatten[t] (* A253281 sequence *)

Cf. A078008, A257569, A253282A253718.

allocated for Clark KimberlingTriangular array: T(h,k) = number of steps from (h,k) to (0,0), where allowable steps are as follows: (x,y) -> (x-(x mod 3), y) if (x mod 3) is not 0, and (x,y) -> (y, x/3) otherwise.

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

1,3

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).

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

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)

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 *)

Cf. A078008, A257569, A253282.

allocated

nonn,tabl,easy

Clark Kimberling, May 02 2015

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allocated for Clark Kimberling

recycled

allocated

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allocated for Eric Chen

allocated

recycled

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allocated for Eric Chen

allocated

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