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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.
1
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
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.
Sequence in context: A054869 A201671 A226590 * A227962 A331105 A255615
KEYWORD
nonn,look,tabf,easy
AUTHOR
Alois P. Heinz, Nov 18 2015
STATUS
approved