|
| |
|
|
A074049
|
|
Tree generated by the Wythoff sequences: a permutation of the positive integers.
|
|
14
| |
|
|
1, 2, 3, 5, 4, 7, 8, 13, 6, 10, 11, 18, 12, 20, 21, 34, 9, 15, 16, 26, 17, 28, 29, 47, 19, 31, 32, 52, 33, 54, 55, 89, 14, 23, 24, 39, 25, 41, 42, 68, 27, 44, 45, 73, 46, 75, 76, 123, 30, 49, 50, 81, 51, 83, 84, 136, 53, 86, 87, 141, 88, 143, 144, 233, 22, 36, 37
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Write t=tau=(1+sqrt(5))/2 and let S be generated by these rules: 1 is in S and if x is in S, then f(x) := [t*x] and g(x) := [(t+1)*x] are in S. Then S is the set of positive integers and the present permutation of S is obtained by arranging S in rows according to the order in which they are generated by f and g, starting with x=1.
The formula indicates the manner in which these numbers arise as a tree: 1 stems to 2, which branches to (3,5), and thereafter, each number branches to a pair:
3->(4,7) and 5->(8,13), etc.
The numbers >1 in the lower Wythoff sequence A000201 occupy the first place in each pair, and the numbers >2 in the upper Wythoff sequence A001950 occupy the second place. The pairs, together with (1,2) are the Wythoff pairs, much studied as the solutions of the Wythoff game. The Wythoff pairs also occur, juxtaposed, in the Wythoff array, A035513.
|
|
|
FORMULA
| Array T(n, k) by rows: T(0, 0)=1; T(1, 0)=2;
T(n, 2j)=Floor(tau*T(n-1, j));
T(n, 2j+1)=Floor((tau+1)*T(n-1, j))
for j=0,1,...,2^(n-1)-1, n>=2.
|
|
|
EXAMPLE
| First levels of the tree:
...................1
...................2
...........3.................5
.......4.......7........8........13
.....6..10...11..18....12..20...21..34
|
|
|
CROSSREFS
| Cf. A074050, A000201, A001950, A035513.
Equals A048680(n-1) + 1.
Sequence in context: A127515 A099424 A117955 * A193973 A127521 A102399
Adjacent sequences: A074046 A074047 A074048 * A074050 A074051 A074052
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Aug 14 2002
|
|
|
EXTENSIONS
| Extended by Clark Kimberling (ck6(AT)evansville.edu), Dec 23 2010
|
| |
|
|
