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%I #9 May 25 2015 15:35:27
%S 1,3,4,10,5,13,14,34,7,17,18,44,19,47,48,116,9,23,24,58,25,61,62,150,
%T 26,64,66,160,67,163,164,396,12,30,32,78,33,81,82,198,35,85,86,208,87,
%U 211,212,512,36,88,90,218,93,225,226,546,94,228
%N First of two trees generated by the Beatty sequence of sqrt(2).
%C This tree grows from (L(1),U(1))=(1,3). The other tree, A183171, grows from (L(2),U(2)=(2,6). Here, L is the Beatty sequence A001951 of r=sqrt(2); U is the Beatty sequence A001952 of s=r/(r-1). The two trees are complementary; that is, every positive integer is in exactly one tree. (L and U are complementary, too.) The sequence formed by taking the terms of this tree in increasing order is A183172.
%H Ivan Neretin, <a href="/A183170/b183170.txt">Table of n, a(n) for n = 1..8192</a>
%F See the formula at A178528, but use r=sqrt(2) instead of r=sqrt(3).
%e First levels of the tree:
%e .......................1
%e .......................3
%e ..............4...................10
%e .........5..........13........14........34
%e .......7..17......18..44....19..47....48..116
%t a = {1, 3}; row = {a[[-1]]}; r = Sqrt[2]; s = r/(r - 1); Do[a = Join[a, row = Flatten[{Floor[#*{r, s}]} & /@ row]], {n, 5}]; a (* _Ivan Neretin_, May 25 2015 *)
%Y Cf. A183171, A183172, A001951, A001952, A178528, A074049.
%K nonn,tabf
%O 1,2
%A _Clark Kimberling_, Dec 28 2010
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