cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A337722 Maximum number of knights within an n X n chessboard, where each knight has a path off the board.

Original entry on oeis.org

1, 4, 9, 16, 24, 34, 44, 58, 73
Offset: 1

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Author

Nicholas Tran, Sep 17 2020

Keywords

Comments

Given an n X n chessboard, this sequence is the maximum number of knights that can be placed on the board, such that each knight has a free path to jump "off" the board without another knight moving.
An example for a(3):
XXX
XXX
XXX
Here, all knights can jump off the board via knight movement. The middle knight is not blocked by any pieces.
An example for a(5):
XXXXX
XXXXX
XX-XX
XXXXX
XXXXX
Here, a knight can occupy all spaces except the central space, where a knight would not be able to jump off the board and would not be able to jump to another free square.

Links

Crossrefs

Cf. A337746 (bishops), A335445 (rooks).

A337746 Maximum number of bishops within an n X n chessboard, where each bishop has a path to an edge.

Original entry on oeis.org

1, 4, 8, 12, 19, 28, 37, 48, 57, 72, 87
Offset: 1

Views

Author

Nicholas Tran, Sep 17 2020

Keywords

Comments

Given an n X n chessboard, this sequence is the maximum number of bishops that can be placed on the board, such that each bishop has a free path to the edge of the board without another bishop moving.
Bishops on even and odd squares don't interfere and can be maximized separately. When n is even, the two types are equal via symmetry and thus the maximum number of bishops will be 2 * (maximum number of one type of bishop).

Links

Crossrefs

Cf. A337722 (knights), A335445 (rooks).

A375762 Maximum number of knights within an n X n chessboard, where each knight has a path to an edge.

Original entry on oeis.org

1, 4, 8, 14, 20, 30, 41, 55
Offset: 1

Views

Author

Walter Robinson, Aug 26 2024

Keywords

Comments

Each knight must be either already on an edge square, or have a path of unoccupied squares which reach an unoccupied edge square (and without any other knights moving).

Examples

			For n=3, the following board, with X for each knight, is the unique solution a(3) = 8 and which cannot be 9 since the central square has no move to anywhere within the board.
  XXX
  X-X
  XXX
For n=4, the following is a solution for a(4) = 14, with each of the 4 central knights able to make a single move to one of the unoccupied corner squares.
  -XX-
  XXXX
  XXXX
  XXXX
For n = 8, one 55 knight solution is:
  XXXXXXXX
  XXXXXXXX
  XX-X-XXX
  XX-X-XXX
  -XX---XX
  XXXX-XXX
  XXXXXXXX
  XXXXXXXX

Crossrefs

Cf. A335445 (rooks), A337746 (bishops), A337722 (knights moving off the board).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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