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.

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

Author

Alois P. Heinz, Jun 14 2023

Keywords

• nonn
• tabf
• look

Comments

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

Examples

			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, ...;
  ...
		

Links

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

Crossrefs

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.

Programs

Maple

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);

Formula

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).