

A074049


Tree generated by the Wythoff sequences: a permutation of the positive integers.


15



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
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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.


LINKS

Ivan Neretin, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers


FORMULA

Array T(n, k) by rows: T(0, 0)=1; T(1, 0)=2;
T(n, 2j) = floor(tau*T(n1, j));
T(n, 2j+1) = floor((tau+1)*T(n1, j))
for j=0,1,...,2^(n1)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


MATHEMATICA

a = {1, 2}; row = {a[[1]]}; r = GoldenRatio; s = r/(r  1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* Ivan Neretin, Nov 09 2015 *)


CROSSREFS

Cf. A074050, A000201, A001950, A035513.
Equals A048680(n1) + 1.
Sequence in context: A256996 A099424 A117955 * A193973 A245057 A127521
Adjacent sequences: A074046 A074047 A074048 * A074050 A074051 A074052


KEYWORD

nonn,tabf


AUTHOR

Clark Kimberling, Aug 14 2002


EXTENSIONS

Extended by Clark Kimberling, Dec 23 2010


STATUS

approved



