A362160 -id:A362160 - cp's OEIS frontend

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

Gary W. Adamson, Sep 23 2003

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)

			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

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

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

Cf. A088372.

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

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

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 *)

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

A363674 T(n,k) is the decimal equivalent of the n-bit inverted Gray code for k; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

Original entry on oeis.org

0, 1, 0, 3, 2, 0, 1, 7, 6, 4, 5, 1, 0, 2, 3, 15, 14, 12, 13, 9, 8, 10, 11, 3, 2, 0, 1, 5, 4, 6, 7, 31, 30, 28, 29, 25, 24, 26, 27, 19, 18, 16, 17, 21, 20, 22, 23, 7, 6, 4, 5, 1, 0, 2, 3, 11, 10, 8, 9, 13, 12, 14, 15, 63, 62, 60, 61, 57, 56, 58, 59, 51, 50, 48

Offset: 0

Alois P. Heinz, Jun 14 2023

Row n is a permutation of {0, 1, ..., A000225(n)}.

			Triangle T(n,k) begins:
   0;
   1,  0;
   3,  2,  0,  1;
   7,  6,  4,  5, 1, 0,  2,  3;
  15, 14, 12, 13, 9, 8, 10, 11, 3, 2, 0, 1, 5, 4, 6, 7;
  ...
T(n,k) written in n-bit binary begins:
    ();
     1,    0;
    11,   10,   00,   01;
   111,  110,  100,  101,  001,  000,  010,  011;
  1111, 1110, 1100, 1101, 1001, 1000, 1010, 1011, 0011, 0010, 0000, ...;
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Wikipedia, Gray code

Columns k=0-2 give: A000225, A000918 (for n>=1), A028399 (for n>=2).
Row sums give A006516.
Cf. A000120, A000975, A003188, A063524, A097072, A236840, A329278, A331105, A362160.

T:= (n, k)-> Bits[Xor](2^n-1-k, iquo(k, 2)):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

T(n,k) = 2^n - 1 - A003188(k) = A000225(n) - A003188(k).
Sum_{k=0..2^n-1} (-1)^k * T(n,k) = A063524(n).
T(n,0) = T(n+1,2^(n+1)-1) = A000225(n).
T(n,A000975(n)) = 0.
T(n,A097072(n)) = 1 for n >= 1.
T(n,k) = T(n-1,k) + 2^(n-1) for n >= 1 and 0 <= k < 2^(n-1).
T(n,k) = T(n-1,2^n-1-k) for n >= 1 and 2^(n-1) <= k < 2^n.
A000120(T(n,n)) = A236840(n).

cp's OEIS Frontend

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.

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