A065775 Array T read by diagonals: T(i,j)=least number of knight's moves on a chessboard (infinite in all directions) needed to move from (0,0) to (i,j).

0, 3, 3, 2, 2, 2, 3, 1, 1, 3, 2, 2, 4, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 2, 2, 2, 4, 4, 5, 3, 3, 3, 3, 3, 3, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 6, 6, 6, 6, 4, 4, 4, 4, 4, 6, 6, 6, 6, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7

Offset: 0

Author

Stewart Gordon, Dec 05 2001

Keywords

• nonn
• tabl
• changed

Examples

			From _Clark Kimberling_, Dec 20 2010: (Start)
T(i,j) for -2<=i<=2 and -2<=j<=2:
  4 1 2 1 4=T(2,2)
  1 2 3 2 1=T(2,1)
  2 3 0 3 2=T(2,0)
  1 2 3 2 1=T(2,-1)
  4 1 2 1 4=T(2,-2)
Corner of the array, T(i,j) for i>=0, j>=0: [Corrected Oct 14 2016]
  0 3 2 3 2 3 4...
  3 2 1 2 3 4 3...
  2 1 4 3 2 3 4...
  3 2 3 2 3 4 2... (End)
		

Crossrefs

Identical to A049604 except for T(1, 1).

For number of knight's moves to various subsets of the chessboard, see A018837, A183041-A183053.

Formula

From Clark Kimberling, Dec 20 2010: (Start)

T(i,j) is given in cases:

Case 1: row 0

T(0,0)=0, T(1,0)=3, and for m>=1,

T(4m-2,0)=2m, T(4m-1,0)=2m+1, T(4m,0)=2m,

T(4m+1,0)=2m+1.

Case 2: row 1

T(0,1)=3, T(1,1)=2, and for m>=1,

T(4m-2,1)=2m-1, T(4m-1,1)=2m, T(4m,1)=2m+1,

T(4m+1,1)=2m+2.

Case 3: columns 0 and 1

(column 0 = row 0); (column 1 = row 1).

Case 4: For i>=2 and j>=2,

T(i,j)=1+min{T(i-2,j-1),T(i-1,j-2)}.

Cases 1-4 determine T in the 1st quadrant;

all other T(i,j) are easily obtained by symmetry. (End)