A322285 -id:A322285 - cp's OEIS frontend

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

Reinhard Zumkeller, Dec 06 2008

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

			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

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

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

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.

A322795 Number of integers k, 0 <= k <= n, such that the Damerau-Levenshtein distance between the binary representations of n and k is strictly less than the Levenshtein distance.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 1, 2, 0, 0, 0, 1, 0, 2, 1, 3, 0, 4, 4, 4, 1, 4, 2, 4, 0, 0, 0, 1, 0, 2, 1, 3, 0, 4, 5, 5, 1, 5, 4, 7, 0, 8, 9, 9, 6, 8, 8, 8, 1, 8, 8, 8, 2, 8, 4, 8, 0, 0, 0, 1, 0, 2, 1, 4, 0, 4, 6, 6, 1, 5, 4, 9, 0, 8, 11, 11, 7, 10, 11, 11, 1, 10, 12, 13, 5, 13, 9, 14, 0, 16, 18, 17, 15, 16

Offset: 0

Pontus von Brömssen, Dec 26 2018

a(n) = 0 if and only if n appears in A099627 or n = 0.

a(n) = A079071(n) for n <= 21, but a(22) = 3 > 2 = A079071(22).

			For n = 6, the Damerau-Levenshtein distance and the Levenshtein distance between the binary representations of n and k are equal for all k <= n except k = 5. The Levenshtein distance between 101 and 110 (5 and 6 in binary) is 2, whereas the Damerau-Levenshtein distance is 1, so a(6) = 1.

Pontus von Brömssen, Table of n, a(n) for n = 0..1000
Wikipedia, Damerau-Levenshtein distance

Cf. A152487, A322285, A099627, A079071.

cp's OEIS Frontend

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

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