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A257569
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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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5
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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MATHEMATICA
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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}];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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