A074049 - OEIS

%I #17 Nov 09 2015 13:11:57

%S 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,

%T 32,52,33,54,55,89,14,23,24,39,25,41,42,68,27,44,45,73,46,75,76,123,

%U 30,49,50,81,51,83,84,136,53,86,87,141,88,143,144,233,22,36,37

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

%C 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.

%C 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:

%C   3->(4,7) and 5->(8,13), etc.

%C 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.

%H Ivan Neretin, <a href="/A074049/b074049.txt">Table of n, a(n) for n = 1..8192</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F Array T(n, k) by rows: T(0, 0)=1; T(1, 0)=2;

%F T(n, 2j) = floor(tau*T(n-1, j));

%F T(n, 2j+1) = floor((tau+1)*T(n-1, j))

%F for j=0,1,...,2^(n-1)-1, n>=2.

%e First levels of the tree:

%e ...................1

%e ...................2

%e ...........3.................5

%e .......4.......7........8........13

%e .....6..10...11..18....12..20...21..34

%t 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 *)

%Y Cf. A074050, A000201, A001950, A035513.

%Y Equals A048680(n-1) + 1.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Aug 14 2002

%E Extended by _Clark Kimberling_, Dec 23 2010