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.

A110214 Minimal number of knights to cover a cubic board.

Original entry on oeis.org

1, 8, 6, 8, 13
Offset: 1

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Author

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jul 17 2005

Keywords

Examples

			Illustration for n = 3, 4, 5 ( O = empty field, K = knight ):
n = 3: OOO KKK OOO n = 4: OOOO OKOO OOOO OOOO
...... OKO OKO OKO ...... OOOO OKKK OOOO OOOO
...... OOO OOO OOO ...... OOOO KKKO OOOO OOOO
......................... OOOO OOKO OOOO OOOO
n = 5: 1, 2, 4 and 5 planes empty, 3 plane: OKOKO OKOKO KKKKK KOKOK OOKOO.

Crossrefs

This is a 3-dimensional version of A006075. a(n) = A110217(n, n, n). A110215 gives number of inequivalent ways to cover the board using a(n) knights, A110216 gives total number.

Formula

Generalize a knight for a spatial board: a move consists of two steps in the first, one step in the second and no step in the third dimension. How many of such knights are needed to occupy or attack every field of an n X n X n board? Knights may attack each other.

A110219 Cone C(n,m,k) read by planes and rows, for 1 <= k <= m <= n: Total Number of coverings of a k X m X n board using A110217(n,m,k) knights.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 36, 8, 12, 12, 1, 16, 1296, 15, 56, 14, 9, 16, 8, 156, 1, 4, 2916, 6, 24, 8, 3, 4, 2, 6, 47, 2, 8, 38, 888, 1, 1, 6561, 2, 236, 2, 1, 268, 1, 2988, 46, 4, 27, 7
Offset: 1

Views

Author

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jul 17 2005

Keywords

Examples

			Cone starts:
1.1...1........1..............1.................1......................
..1,1.4,36....16,1296.........4,2916............1,6561.
......8,12,12.15,..56,14......6,..24,8..........2,.236,.2
...............9,..16,.8,156..3,...4,2,.6.......1,.268,.1,2988
.............................47,...2,8,38,888..46,...4,27,...7,.?
..............................................127,..32,12,...?,....

Crossrefs

C(n, n, 1) = A103315(n), C(n, n, n) = A110216(n). A110218 gives number of inequivalent solutions.

A110215 Inequivalent coverings of a cubic board with the minimal number of knights.

Original entry on oeis.org

1, 1, 1, 8, 37
Offset: 1

Views

Author

Nikolaus Meyberg (Nikolaus.Meyberg(AT)t-online.de), Jul 17 2005

Keywords

Comments

This sequence is a 3-dimensional analog of A006076.
Sequence A110214 gives minimal number of knights needed to cover an n X n X n board. This sequence gives the number of inequivalent solutions to cover an n X n X n board using A110214(n) knights.

Examples

			a(3) = 1, since up to rotations and reflections,
OOO KKK OOO
OKO OKO OKO
OOO OOO OOO is the only covering for n = 3.

Crossrefs

Cf. A006076, A110214, A110218.
A110216 gives total number of solutions.

Formula

a(n) = A110218(n, n, n).
Showing 1-3 of 3 results.

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