A152487 Triangle read by rows, 0<=k<=n: T(n,k) = Levenshtein distance of n and k in binary representation.

0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 2, 0, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 1, 1, 0, 3, 3, 2, 3, 1, 2, 2, 3, 0, 3, 3, 2, 2, 1, 1, 2, 2, 1, 0, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 0, 3, 3, 2, 2, 2, 1, 2, 1, 2, 1, 1, 0, 3, 3, 2, 2, 1, 2, 1, 2, 1, 2, 2, 3, 0, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 0

Offset: 0

Author

Reinhard Zumkeller, Dec 06 2008

Keywords

• nonn
• base
• tabl

Comments

T(n,k) gives number of editing steps (replace, delete and insert) to transform n to k in binary representations;

row sums give A152488; central terms give A057427;

T(n,k) <= Hamming-distance(n,k) for n and k with A070939(n)=A070939(k);

T(n,0) = A000523(n+1);

T(n,1) = A000523(n) for n>0;

T(n,3) = A106348(n-2) for n>2;

T(n,n-1) = A091090(n-1) for n>0;

T(n,n) = A000004(n);

T(A000290(n),n) = A091092(n).

T(n,k) >= A322285(n,k) - Pontus von Brömssen, Dec 02 2018

Examples

			The triangle T(n, k) begins:
  n\k  0  1  2  3  4  5  6  7  8  9 10 11 12 13 ...
   0:  0
   1:  1  0
   2:  1  1  0
   3:  2  1  1  0
   4:  2  2  1  2  0
   5:  2  2  1  1  1  0
   6:  2  2  1  1  1  2  0
   7:  3  2  2  1  2  1  1  0
   8:  3  3  2  3  1  2  2  3  0
   9:  3  3  2  2  1  1  2  2  1  0
  10:  3  3  2  2  1  1  1  2  1  2  0
  11:  3  3  2  2  2  1  2  1  2  1  1  0
  12:  3  3  2  2  1  2  1  2  1  2  2  3  0
  13:  3  3  2  2  2  1  1  1  2  1  2  2  1  0
  ...
The distance between the binary representations of 46 and 25 is 4 (via the edits "101110" - "10111" - "10011" - "11011" - "11001"), so T(46,25) = 4. - _Pontus von Brömssen_, Dec 02 2018
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Michael Gilleland, Levenshtein Distance
• Wikipedia, Levenshtein Distance
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000004, A000290, A000523, A057427, A070939, A091090, A091092, A106348, A152488, A322285.

Formula

T(n,k) = f(n,k) with f(x,y) = if x>y then f(y,x) else if x<=1 then Log2(y)-0^y+(1-x)*0^(y+1-2^(y+1)) else Min{f([x/2],[y/2]) + (x mod 2) XOR (y mod 2), f([x/2],y)+1, f(x,[y/2])+1}, where Log2=A000523.