A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2*n+1) X (2*n+1) board starting from the k-th outermost square counted from middle of first rank for k = 1..n+1, or -1 if reaching the central square is impossible.

0, -1, -1, 4, 1, 2, 2, 3, 2, 3, 4, 3, 2, 3, 2, 4, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 6, 5, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 4, 5, 4, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 8, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6

Offset: 0

Author

Felix Fröhlich, Dec 15 2016

Keywords

• sign
• look
• tabl

Examples

			Triangle starts
   0;
  -1, -1;
   4,  1,  2;
   2,  3,  2,  3;
   4,  3,  2,  3,  2;
   4,  3,  4,  3,  4,  3;
   4,  5,  4,  3,  4,  3,  4;
   6,  5,  4,  5,  4,  5,  4,  5;
   6,  5,  6,  5,  4,  5,  4,  5,  4;
   6,  7,  6,  5,  6,  5,  6,  5,  6,  5;
   ...
T(0, 1) = 0, because the board has just 1 square where the knight must start.
T(1, 1) and T(1, 2) = -1, because reaching the central square with a knight is not possible on a 3 X 3 board.
T(2, 1) = 4, because at least 4 moves are necessary on a 5 X 5 board to reach the central square when starting from a corner square.
T(2, 3) = 2 because 2 moves are necessary on a 5 X 5 board to reach the central square when starting from the middle of one side. - _Andrew Howroyd_, Feb 28 2020
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows n = 0..50)
• Wikipedia, Jeson Mor.

Crossrefs

Cf. A035008, A049604, A065775.

Formula

T(n,k) = A049604(n, n-k) = A065775(n, n-k) for n > 1. - Andrew Howroyd, Feb 28 2020

Extensions

a(5) corrected and terms a(15) and beyond from Andrew Howroyd, Feb 28 2020