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

0,5

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.

Links

Alois P. Heinz, Rows n = 0..13, flattened

Wikipedia, Gray code

Wikipedia, Hamming distance

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).

Example

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, ... ;

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);

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.

Sequence in context: A054869 A201671 A226590 * A227962 A331105 A255615

Adjacent sequences: A261346 A261347 A261348 * A261350 A261351 A261352

Keyword

nonn,look,tabf,easy

Author

Alois P. Heinz, Nov 18 2015

Status

approved