A261349 T(n,k) is the decimal equivalent of a code for k that maximizes the sum of the Hamming distances between (cyclical) adjacent code words; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.

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

Offset: 0

Author

Alois P. Heinz, Nov 18 2015

Keywords

• nonn
• look
• tabf
• easy

Comments

This code might be called "Anti-Gray code".

The sum of the Hamming distances between (cyclical) adjacent code words of row n gives 0, 2, 6, 20, 56, 144, 352, ... = A014480(n-1) for n>1.

Examples

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  3, 1,  2;
  0,  7, 1,  6, 3,  4, 2,  5;
  0, 15, 1, 14, 3, 12, 2, 13, 6,  9, 7,  8, 5, 10, 4, 11;
  0, 31, 1, 30, 3, 28, 2, 29, 6, 25, 7, 24, 5, 26, 4, 27, 12, 19, ... ;
  0, 63, 1, 62, 3, 60, 2, 61, 6, 57, 7, 56, 5, 58, 4, 59, 12, 51, ... ;
		

Links

• Alois P. Heinz, Rows n = 0..13, flattened
• Wikipedia, Gray code
• Wikipedia, Hamming distance

Crossrefs

Columns k=0-3 give: A000004, A000225, A000012 (for n>1), A000918 (for n>1).

Row lengths give A000079.

Row sums give A006516.

Cf. A003188, A014480, A083329, A083420, A101080.

Programs

Maple

g:= n-> Bits[Xor](n, iquo(n, 2)):
T:= (n, k)-> (t-> `if`(m=0, t, 2^n-1-t))(g(iquo(k, 2, 'm'))):
seq(seq(T(n, k), k=0..2^n-1), n=0..6);

Formula

T(n,k) = A003188(k/2) if k even, T(n,k) = 2^n-1-A003188((k-1)/2) else.

A101080(T(n,2k),T(n,2k+1)) = n, A101080(T(n,2k),T(n,2k-1)) = n-1.

T(n,2^n-1) = A083329(n-1) for n>0.

T(n,2^n-2) = A000079(n-2) for n>1.

T(2n,2n) = A003188(n).

T(2n+1,2n+1) = 2*4^n - 1 - A003188(n) = A083420(n) - A003188(n).