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A088208 Table read by rows where T(0,0)=1; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. For j==0 mod 2, T(n+1,2j)=T(n,j) and T(n+1,2j+1)=T(n,j)+2^n. For j==1 mod 2, T(n+1,2j+1)=T(n,j) and T(n+1,2j)=T(n,j)+2^n.

%I A088208 #33 Dec 08 2024 02:26:09
%S A088208 1,1,2,1,3,4,2,1,5,7,3,4,8,6,2,1,9,13,5,7,15,11,3,4,12,16,8,6,14,10,2,
%T A088208 1,17,25,9,13,29,21,5,7,23,31,15,11,27,19,3,4,20,28,12,16,32,24,8,6,
%U A088208 22,30,14,10,26,18,2,1,33,49,17,25,57,41,9,13,45,61,29,21,53,37,5,7,39,55,23
%N A088208 Table read by rows where T(0,0)=1; n-th row has 2^n terms T(n,j),j=0 to 2^n-1. For j==0 mod 2, T(n+1,2j)=T(n,j) and T(n+1,2j+1)=T(n,j)+2^n. For j==1 mod 2, T(n+1,2j+1)=T(n,j) and T(n+1,2j)=T(n,j)+2^n.
%C A088208 Schroeder, p. 281 states "The ordering with which the iterates x_n fall into the 2^m different chaos bands [order as to magnitude] is also the same as the ordering of the iterates in a stable orbit of period length P = 2^m. For example, for both the period-4 orbit and the four chaos bands, the iterates, starting with the largest iterate x_1, are ordered as follows: x_1 > x_3 > x_4 > x_2."
%C A088208 From _Andrey Zabolotskiy_, Dec 06 2024: (Start)
%C A088208 For n>0, row n-1 is the permutation relating row n of the left half of Stern-Brocot tree with row n of Kepler's tree of fractions. Specifically, if K_n(k) [resp. SB_n(k)] is the k-th fraction in the n-th row of A294442 [resp. A057432], where 1/2 is in row 1 and k=1..2^(n-1), then SB_n(k) = K_n(T(n-1, k)).
%C A088208 The inverse permutation is row n of A131271.
%C A088208 Equals A362160+1. (End)
%D A088208 Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W.H. Freeman, 1991, p. 282.
%H A088208 Reinhard Zumkeller, <a href="/A088208/b088208.txt">Rows n = 1..13 of triangle, flattened</a>
%e A088208 1
%e A088208 1 2
%e A088208 1 3 4 2
%e A088208 1 5 7 3 4 8 6 2
%e A088208 1 9 13 5 7 15 11 3 4 12 16 8 6 14 10 2
%t A088208 nmax = 6;
%t A088208 T[_, 0] = 1; T[n_, j_] /; j == 2^n = n;
%t A088208 Do[Which[
%t A088208   EvenQ[j], T[n+1, 2j] = T[n, j]; T[n+1, 2j+1] = T[n, j] + 2^n,
%t A088208   OddQ[j], T[n+1, 2j+1] = T[n, j]; T[n+1, 2j] = T[n, j] + 2^n],
%t A088208 {n, 0, nmax}, {j, 0, 2^n-1}];
%t A088208 Table[T[n, j], {n, 0, nmax}, {j, 0, 2^n-1}] // Flatten (* _Jean-François Alcover_, Aug 03 2018 *)
%o A088208 (Haskell)
%o A088208 a088208 n k = a088208_tabf !! (n-1) !! (k-1)
%o A088208 a088208_row n = a088208_tabf !! (n-1)
%o A088208 a088208_tabf = iterate f [1] where
%o A088208    f vs = (map (subtract 1) ws) ++ reverse ws where ws = map (* 2) vs
%o A088208 -- _Reinhard Zumkeller_, Mar 14 2015
%Y A088208 Cf. A088372.
%Y A088208 Cf. A049773, A362160, A131271, A294442, A057432.
%K A088208 nonn,tabf
%O A088208 1,3
%A A088208 _Gary W. Adamson_, Sep 23 2003
%E A088208 Edited by _Ray Chandler_ and _N. J. A. Sloane_, Oct 08 2003