A065775 -id:A065775 - cp's OEIS frontend

A183046 Sums of knight's moves from (0,0) to vertical segments (n,-n) to (n,n) on infinite chessboard.

0, 7, 12, 17, 26, 39, 50, 71, 84, 109, 128, 157, 178, 215, 238, 279, 308, 353, 384, 437, 470, 527, 566, 627, 668, 737, 780, 853, 902, 979, 1030, 1115, 1168, 1257, 1316, 1409, 1470, 1571, 1634, 1739, 1808, 1917, 1988, 2105

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

Cf.A065775, A183045.

T(n,-n)+T(n,-n+1)+...+T(n,n), where T is given at A065775.

Empirical g.f.: x*(4*x^8+2*x^7-8*x^6-7*x^5+5*x^3-3*x^2-12*x-7) / ((x-1)^3*(x+1)^2*(x^2+x+1)). - Colin Barker, May 04 2014

A183047 Sums of least knight's moves from (0,0) to points in the square lattice [-n,n]x[-n,n].

Original entry on oeis.org

0, 20, 52, 112, 200, 340, 524, 784, 1096, 1508, 1988, 2584, 3264, 4084, 4996, 6072, 7256, 8620, 10108, 11800, 13624, 15676, 17876, 20320, 22928, 25804, 28852, 32192, 35720, 39556, 43596, 47968, 52552, 57492, 62660, 68200, 73984

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

			a(2)=52 counts the knight's moves to these points:
4 1 2 1 4
1 2 3 2 1
2 3 0 3 2
1 2 3 2 1
4 1 2 1 4
a(0)=0 for the center, and a(1)=20 for the square
2 3 2
3 0 3
2 3 2

Cf. A065775, A183048.

See A065775.

Empirical g.f.: -4*x*(2*x^8+2*x^7-4*x^6-5*x^5-2*x^4-x^3-5*x^2-8*x-5) / ((x-1)^4*(x+1)^2*(x^2+x+1)). - Colin Barker, May 04 2014

A183048 Sums of least number of knight's moves on boundaries of squares [-n,n]x[-n,n] on infinite chessboard.

Original entry on oeis.org

0, 20, 32, 60, 88, 140, 184, 260, 312, 412, 480, 596, 680, 820, 912, 1076, 1184, 1364, 1488, 1692, 1824, 2052, 2200, 2444, 2608, 2876, 3048, 3340, 3528, 3836, 4040, 4372, 4584, 4940, 5168, 5540, 5784, 6180, 6432

Offset: 0

Clark Kimberling, Dec 20 2010

nonn

First difference sequence of A183047.

Every term is divisible by 4.

			Start with the square [-2,2]x[2,2],
4 1 2 1 4
1 2 3 2 1
2 3 0 3 2
1 2 3 2 1
4 1 2 1 4,
remove the square [-1,1]x[-1,1],
2 3 4
3 0 3
2 3 2,
and then add the remaining numbers:
4+1+2+1+4+1+2+1+4+1+2+1+4+1+2+1
to get a(2)=32.

Cf. A065775, A183047.

See A065775.

Empirical g.f.: 4*x*(2*x^8+2*x^7-4*x^6-5*x^5-2*x^4-x^3-5*x^2-8*x-5) / ((x-1)^3*(x+1)^2*(x^2+x+1)). - Colin Barker, May 04 2014

Duplicate term 820 deleted by Colin Barker, Feb 19 2014

A183049 Array of least knight's moves to points (n,0), (n-1,1), ..., (1,n-1) on infinite chessboard.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 22 2010

The n points (n,0), (n-1,1), ..., (1,n-1) lie in a diagonal in the first quadrant. Adjoining the matching points in the other quadrants yields the square |i|+|j|=n, as in A183051. For a description of the infinite chessboard, see A065775.

			First 6 rows (after the initial 0):
3
2 2
3 1 1
2 2 4 2
3 3 3 3 3
4 4 2 2 2 4
These numbers occupy positions on the chessboard as
indicated here, starting at the left bottom corner:
..4
..3 4
..2 3 2
..1 4 3 2
..2 1 2 3 4
0 3 2 3 2 3 4 ... (This row is A018837.)

Cf. A065775, A183050, A183051.

See A065775.

A246924 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the square (1,2).

Original entry on oeis.org

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

Offset: 1

Frieder Mittmann, Sep 07 2014

Cf. A018837, A246925, A065775.

A246925 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the corner square (1,1).

Original entry on oeis.org

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

Offset: 1

Frieder Mittmann, Sep 07 2014

The initial position needs no moves to be reached and is set to value 0
Read by rows:
a(1)   a(2) .... a(8)
a(9)  a(10) .... a(16)
   ....
   ....
a(57) a(58) .... a(64)

Cf. A018837, A246924. See A065775 for the analog on an infinite board.

A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2n+1) X (2n+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.

Original entry on oeis.org

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

Felix Fröhlich, Dec 15 2016

			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

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

Cf. A035008, A049604, A065775.

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

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

cp's OEIS Frontend

A183046 Sums of knight's moves from (0,0) to vertical segments (n,-n) to (n,n) on infinite chessboard.

Views

Author

Keywords

Crossrefs

Formula

A183047 Sums of least knight's moves from (0,0) to points in the square lattice [-n,n]x[-n,n].

Views

Author

Keywords

Examples

Crossrefs

Formula

A183048 Sums of least number of knight's moves on boundaries of squares [-n,n]x[-n,n] on infinite chessboard.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A183049 Array of least knight's moves to points (n,0), (n-1,1), ..., (1,n-1) on infinite chessboard.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A246924 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the square (1,2).

Views

Author

Keywords

Crossrefs

A246925 8 X 8 square array read by rows: T(i,j) = number of moves of a knight to reach the square (i,j) when starting from the corner square (1,1).

Views

Author

Keywords

Comments

Crossrefs

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A279605 Triangle T(n, k) read by rows: minimal number of knight moves to reach the central square on a (2n+1) X (2n+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.