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

0,7

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

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

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.

Example

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

Crossrefs

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

Sequence in context: A152146 A025860 A322285 * A356894 A338019 A058394

Adjacent sequences: A152484 A152485 A152486 * A152488 A152489 A152490

Keyword

nonn,base,tabl

Author

Reinhard Zumkeller, Dec 06 2008

Status

approved