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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.

%I A257569 #6 May 02 2015 10:14:57
%S A257569 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,
%T A257569 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,
%U A257569 9,9,9,9,9,9,9,10,10,10,10,8,10,9,10,8
%N 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.
%C A257569 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).
%C A257569 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.
%C A257569 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).
%C A257569 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.
%H A257569 Clark Kimberling, <a href="/A257569/b257569.txt">Table of n, a(n) for n = 1..1000</a>
%e A257569 First ten rows:
%e A257569 0
%e A257569 1   2
%e A257569 3   3   4
%e A257569 4   4   5   5
%e A257569 5   5   5   6   6
%e A257569 6   6   6   6   7   7
%e A257569 6   7   6   7   7   8   7
%e A257569 7   7   7   7   8   8   8   8
%e A257569 7   8   7   8   7   9   8   9   8
%e A257569 8   8   8   8   8   8   9   9   9   9
%e A257569 Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths to (0,0) are as shown here:
%e A257569 (2,0) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
%e A257569 (1,1) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
%e A257569 (0,2) -> (2,0) -> (0,1) -> (1,0) -> (0,0) (4 steps).
%t A257569 f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
%t A257569 g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
%t A257569 h[{x_, y_}] := -1 + Length[g[{x, y}]];
%t A257569 t = Table[h[{n - k, k}], {n, 0, 16}, {k, 0, n}];
%t A257569 TableForm[t] (* A257569 array *)
%t A257569 Flatten[t]   (* A257569 sequence *)
%Y A257569 Cf. A257570, A257571, A257572, A000045, A000931, A029744.
%K A257569 nonn,tabl,easy
%O A257569 1,3
%A A257569 _Clark Kimberling_, May 01 2015