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A074049 Tree generated by the Wythoff sequences: a permutation of the positive integers.

%I A074049 #17 Nov 09 2015 13:11:57
%S A074049 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 A074049 32,52,33,54,55,89,14,23,24,39,25,41,42,68,27,44,45,73,46,75,76,123,
%U A074049 30,49,50,81,51,83,84,136,53,86,87,141,88,143,144,233,22,36,37
%N A074049 Tree generated by the Wythoff sequences: a permutation of the positive integers.
%C A074049 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 A074049 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 A074049   3->(4,7) and 5->(8,13), etc.
%C A074049 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 A074049 Ivan Neretin, <a href="/A074049/b074049.txt">Table of n, a(n) for n = 1..8192</a>
%H A074049 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A074049 Array T(n, k) by rows: T(0, 0)=1; T(1, 0)=2;
%F A074049 T(n, 2j) = floor(tau*T(n-1, j));
%F A074049 T(n, 2j+1) = floor((tau+1)*T(n-1, j))
%F A074049 for j=0,1,...,2^(n-1)-1, n>=2.
%e A074049 First levels of the tree:
%e A074049 ...................1
%e A074049 ...................2
%e A074049 ...........3.................5
%e A074049 .......4.......7........8........13
%e A074049 .....6..10...11..18....12..20...21..34
%t A074049 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 A074049 Cf. A074050, A000201, A001950, A035513.
%Y A074049 Equals A048680(n-1) + 1.
%K A074049 nonn,tabf
%O A074049 1,2
%A A074049 _Clark Kimberling_, Aug 14 2002
%E A074049 Extended by _Clark Kimberling_, Dec 23 2010