A091092 -id:A091092 - 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.

A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

Original entry on oeis.org

0, 0, 2, 1, 1, 2, 3, 2, 3, 2, 2, 4, 2, 4, 3, 3, 4, 4, 4, 4, 5, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 3, 3, 5, 3, 5, 3, 3, 5, 5, 5, 6, 4, 3, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 4, 5, 4, 4, 5, 5, 4, 5, 4, 5, 5, 5, 4, 6, 4, 5, 4, 5, 4, 5, 4, 4, 5, 4, 4, 4, 5, 5, 5, 4, 4, 6, 4, 5, 4, 4, 5, 5, 5, 5, 6

Offset: 0

Reinhard Zumkeller, Dec 18 2003

			a(12)=2: 12->'110', insert a 2 between the 1's and insert a 0 at the end: '12100'->144=12^2.

Robert Israel, Table of n, a(n) for n = 0..10000
Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A091092, A091091, A081604, A000290.

A091093:= proc(x) local L1, L2;
   L1:= convert(map(`+`,ListTools:-Reverse(convert(x,base,3)),48),bytes);
   L2:= convert(map(`+`,ListTools:-Reverse(convert(x^2,base,3)),48),bytes);
   StringTools:-Levenshtein(L1,L2)
end proc:
seq(A091093(i),i=0..1000); # Robert Israel, May 06 2014

a(n) = LevenshteinDistance(A007089(n), A001738(n)).

A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

Original entry on oeis.org

0, 1, 5, 6, 8, 11, 13, 16, 17, 18, 19, 33, 35, 38, 40, 56, 60, 74, 122, 123, 133, 143, 146, 164, 168, 173, 299, 350, 365, 429, 497, 515, 527, 564, 566, 593, 608, 611, 710, 1031, 1050, 1052, 1059, 1088, 1089, 1090, 1092, 1096, 1105, 1287, 1301, 1316, 1322

Offset: 1

Reinhard Zumkeller, Dec 18 2003

A091092(a(n)) = A091093(a(n)).

Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Binary
Eric Weisstein's World of Mathematics, Ternary

Cf. A007088, A001737, A007089, A001738, A000290.

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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A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

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A091091 Numbers needing in binary and ternary representation an equal minimal number of editing steps (delete, insert or substitute) to transform them into their square.

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Author

Keywords

Comments

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Crossrefs