A257571 -id:A257571 - cp's OEIS frontend

A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

0, 1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 6, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 7, 8, 7, 8, 7, 9, 8, 9, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 8, 9, 8, 9, 8, 9, 8, 10, 9, 10, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 8, 10, 9, 10, 8

Offset: 1

Clark Kimberling, May 01 2015

The number of pairs (h,k) satisfying T(h,k) = n is F(n), where F = A000045, the Fibonacci numbers.  The number of such pairs having odd h is F(n-2), and the number having even h is F(n-1).
Let c(n,k) be the number of pairs (h,k) satisfying T(h,k) = n; in particular, c(n,0) is the number of integers (pairs of the form (h,0)) satisfying T(h,0) = n.  Let p(n) = A000931(n).  Then c(n,0) = p(n+3) for n >= 0.  More generally, for fixed k >=0, the sequence satisfies the recurrence r(n) = r(n-2) + r(n-3) except for initial terms.
The greatest h for which some (h,k) is n steps from (0,0) is H = A029744(n-1) for n >= 2, and the only such pair is (H,0).
T(n,k) is also the number of steps from h + k*sqrt(2) to 0, where one step is x + y*sqrt(2) -> x-1 + y*sqrt(2) if x is odd, and x + sqrt(y) -> y + (x/2)*sqrt(2) if x is even.

			First ten rows:
0
1   2
3   3   4
4   4   5   5
5   5   5   6   6
6   6   6   6   7   7
6   7   6   7   7   8   7
7   7   7   7   8   8   8   8
7   8   7   8   7   9   8   9   8
8   8   8   8   8   8   9   9   9   9
Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths to (0,0) are as shown here:
(2,0) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(1,1) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(0,2) -> (2,0) -> (0,1) -> (1,0) -> (0,0) (4 steps).

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

Cf. A257570, A257571, A257572, A000045, A000931, A029744.

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, 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, 16}, {k, 0, n}];
TableForm[t] (* A257569 array *)
Flatten[t]   (* A257569 sequence *)

A257570 Rectangular array, read by antidiagonals: d(h,k) = distance between h and k in the tree at A232558, for h >=0, k >= 0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 01 2015

The distance between h and k is the length of the path from h to k in the tree defined from the root 0 by edges from x to x+1 and x to 2x if x is even, and an edge from x to x+1 if x is odd. This is the tree defined at A232558; it is a subtree of the tree defined at A257569.

			Northwest corner:
0  1  2  3  3  4  4  5  4  5  5
1  0  1  2  2  3  3  4  3  4  4
2  1  0  1  1  2  2  3  2  3  3
3  2  1  0  2  3  1  2  3  4  4
3  2  1  2  0  1  3  4  1  2  2
4  3  2  3  1  0  4  5  2  3  1
d(4,6) = d(6,4) = 3 counts the edges in the path 6,3,2,4;
d(46,21) = 6 counts the edges in the path 46,23,22,11,10,20,21.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A257571, A232558.

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
s[n_] := Reverse[Select[Sort[Flatten[Select[g[{n, 0}], #[[2]] == 0 &]]], # > 0 &]];
m[h_, k_] := Max[Intersection[s[h], s[k]]];
j[h_, k_] := Join[Select[s[h], # >= m[h, k] &], Reverse[Select[s[k], # > m[h, k] &]]];
d[h_, k_] := If[k*h == 0, Length[j[h, k]], -1 + Length[j[h, k]]];
TableForm[Table[d[h, k], {h, 0, 16}, {k, 0, 16}]]  (* A257570 array *)
Flatten[Table[d[h - k, k], {h, 0, 20}, {k, 0, h}]  (* A257570 sequence *)]

cp's OEIS Frontend

A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

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