%I #17 Dec 26 2018 16:57:01

%S 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,

%T 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,

%U 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

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

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

%C row sums give A152488; central terms give A057427;

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

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

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

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

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

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

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

%C T(n,k) >= A322285(n,k) - _Pontus von Brömssen_, Dec 02 2018

%H Alois P. Heinz, <a href="/A152487/b152487.txt">Rows n = 0..200, flattened</a>

%H Michael Gilleland, <a href="http://people.cs.pitt.edu/~kirk/cs1501/Pruhs/Fall2006/Assignments/editdistance/Levenshtein%20Distance.htm">Levenshtein Distance</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Levenshtein_distance">Levenshtein Distance</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F 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.

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ...

%e 0: 0

%e 1: 1 0

%e 2: 1 1 0

%e 3: 2 1 1 0

%e 4: 2 2 1 2 0

%e 5: 2 2 1 1 1 0

%e 6: 2 2 1 1 1 2 0

%e 7: 3 2 2 1 2 1 1 0

%e 8: 3 3 2 3 1 2 2 3 0

%e 9: 3 3 2 2 1 1 2 2 1 0

%e 10: 3 3 2 2 1 1 1 2 1 2 0

%e 11: 3 3 2 2 2 1 2 1 2 1 1 0

%e 12: 3 3 2 2 1 2 1 2 1 2 2 3 0

%e 13: 3 3 2 2 2 1 1 1 2 1 2 2 1 0

%e ...

%e 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

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

%K nonn,base,tabl

%O 0,7

%A _Reinhard Zumkeller_, Dec 06 2008