A253272 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, except that (2,0) -> (1,0).

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

Offset: 1

Author

Clark Kimberling, May 01 2015

Keywords

• nonn
• tabl
• easy

Comments

For n>=3, the number of pairs (h,k) satisfying T(h,k) = n is L(n-1), where L = A000032, the Lucas numbers. The number of such pairs having odd n is L(n-3) for n >= 4, and the number having even n is L(n-2) for n >= 3.

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 >= 2. 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) for n >= 2, and the only such pair is (H,0).

See A257569 for a very similar array for which the number of pairs (h,k) satisfying T(h,k) = n is F(n), where F = A000045, the Fibonacci numbers.

Examples

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

Links

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

Crossrefs

Cf. A257569, A000032, A000931, A029744.

Programs

Mathematica

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}]; f[{2, 0}] = {1, 0};
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]   (* A253272 array *)
u = Flatten[t] (* A253272 sequence *)