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.

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Sep 23 2003

Keywords

• nonn
• tabf

Comments

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."

From Andrey Zabolotskiy, Dec 06 2024: (Start)

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)).

The inverse permutation is row n of A131271.

Equals A362160+1. (End)

Examples

			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
		

References

• Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W.H. Freeman, 1991, p. 282.

Links

• Reinhard Zumkeller, Rows n = 1..13 of triangle, flattened

Crossrefs

Cf. A088372.

Cf. A049773, A362160, A131271, A294442, A057432.

Programs

Haskell

a088208 n k = a088208_tabf !! (n-1) !! (k-1)
a088208_row n = a088208_tabf !! (n-1)
a088208_tabf = iterate f [1] where
   f vs = (map (subtract 1) ws) ++ reverse ws where ws = map (* 2) vs
-- Reinhard Zumkeller, Mar 14 2015

Mathematica

nmax = 6;
T[, 0] = 1; T[n, j_] /; j == 2^n = n;
Do[Which[
  EvenQ[j], T[n+1, 2j] = T[n, j]; T[n+1, 2j+1] = T[n, j] + 2^n,
  OddQ[j], T[n+1, 2j+1] = T[n, j]; T[n+1, 2j] = T[n, j] + 2^n],
{n, 0, nmax}, {j, 0, 2^n-1}];
Table[T[n, j], {n, 0, nmax}, {j, 0, 2^n-1}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

Extensions

Edited by Ray Chandler and N. J. A. Sloane, Oct 08 2003