

A257569


Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) > (x1, y) if x is odd or (x,y) > (y, x/2) if x is even.


5



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

1,3


COMMENTS

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(n2), and the number having even h is F(n1).
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(n2) + r(n3) except for initial terms.
The greatest h for which some (h,k) is n steps from (0,0) is H = A029744(n1) 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) > x1 + y*sqrt(2) if x is odd, and x + sqrt(y) > y + (x/2)*sqrt(2) if x is even.


LINKS

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


EXAMPLE

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).


MATHEMATICA

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 *)


CROSSREFS

Cf. A257570, A257571, A257572, A000045, A000931, A029744.
Sequence in context: A072649 A266082 A105195 * A039836 A083398 A221671
Adjacent sequences: A257566 A257567 A257568 * A257570 A257571 A257572


KEYWORD

nonn,tabl,easy


AUTHOR

Clark Kimberling, May 01 2015


STATUS

approved



