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A152487 Triangle read by rows, 0<=k<=n: T(n,k) = Levenshtein distance of n and k in binary representation. 10
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 (list; table; graph; refs; listen; history; text; internal format)
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
Michael Gilleland, Levenshtein Distance
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
Sequence in context: A152146 A025860 A322285 * A356894 A338019 A058394
KEYWORD
nonn,base,tabl
AUTHOR
Reinhard Zumkeller, Dec 06 2008
STATUS
approved

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)