login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 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 (list; table; graph; refs; listen; history; text; internal format)
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(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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 25 19:24 EDT 2017. Contains 284082 sequences.