A018807 -id:A018807 - cp's OEIS frontend

A195650 Number of ways to place 11n nonattacking kings on a 22 x 2n chessboard.

24576, 18174084, 3449705600, 317076250136, 18589546217696, 803491953235264, 27889602664055396, 821112680030028632, 21279238303065874504, 498306336520679626558, 10749154284380665611224, 216711725342137199240416, 4129262403388762742636600

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

Alex V. Breger, Table of n, a(n) for n = 1..500
Vaclav Kotesovec, Non-attacking chess pieces
Vaclav Kotesovec, Denominator of the generating function
Index entries for linear recurrences with constant coefficients, order 2321.

Column k=11 of A350819.

Cf. A018807, A195655, A195660.

Recurrence order is 2321.

A195651 Number of ways to place 12n nonattacking kings on a 24 x 2n chessboard.

Original entry on oeis.org

53248, 61892669, 16333065216, 1955475353217, 143422674213726, 7545414941610145, 312546900470579954, 10819171744710664383, 325878859655043000344, 8788579757709800395287, 216711725342137199240416, 4963704194366362387891227, 106899958975789427315593702

Offset: 1

Vaclav Kotesovec, Sep 22 2011

nonn

Alex V. Breger, Table of n, a(n) for n = 1..500
Vaclav Kotesovec, Non-attacking chess pieces
Vaclav Kotesovec, Denominator of the generating function
Index entries for linear recurrences with constant coefficients, order 5672.

Column k=12 of A350819.

Cf. A018807, A195656, A195661.

Recurrence order is 5672.

A319096 Number of nonequivalent ways to place n^2 nonattacking kings on a 2n X 2n chessboard under all symmetry operations of the square.

Original entry on oeis.org

1, 14, 459, 35312, 4072108, 638653285, 128441726634, 31872148398195, 9490641145219266, 3321018871480028710

Offset: 1

Anton Nikonov, Dec 21 2018

A maximum of n^2 nonattacking kings may be placed on a 2n X 2n chessboard.

			For n = 2 there are a(2) = 14 distinct solutions from 79 that will not be repeated at all possible turns and reflections.
------------
1.                  2.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
| * |   | * |   |   | * |   |   | * |
|   |   |   |   |   |   |   |   |   |
------------
3.                  4.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
| * |   |   |   |   |   | * |   | * |
|   |   |   | * |   |   |   |   |   |
------------
5.                  6.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   | * |   |   |   |   |   | * |   |
|   |   |   | * |   | * |   |   |   |
------------
7.                  8.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   |   |   | * |   |   |   |   |   |
| * |   |   |   |   | * |   | * |   |
------------
9.                  10.
_________________   _________________
| * |   | * |   |   | * |   | * |   |
|   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |   | * |
| * |   |   | * |   |   | * |   |   |
------------
11.                 12.
_________________   _________________
| * |   | * |   |   | * |   |   | * |
|   |   |   |   |   |   |   |   |   |
|   |   |   |   |   |   | * |   |   |
|   | * |   | * |   |   |   |   | * |
------------
13.                 14.
_________________   _________________
| * |   |   | * |   |   | * |   |   |
|   |   |   |   |   |   |   |   | * |
|   |   |   |   |   | * |   |   |   |
| * |   |   | * |   |   |   | * |   |
------------

Cf. A018807 (rotations and reflections considered distinct).
Cf. A137432 (on cylindrical chessboard).
Cf. A236679, A322284, A321614.

a(n) = A236679(2n+1, n^2).

a(4)-a(10) from Andrew Howroyd, Dec 21 2018

A350818 Array read by antidiagonals: T(m,n) is the number of maximum independent sets in the m X n king graph.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 12, 1, 12, 1, 1, 1, 4, 8, 9, 9, 8, 4, 1, 1, 1, 32, 1, 79, 1, 32, 1, 1, 1, 5, 16, 16, 27, 27, 16, 16, 5, 1, 1, 1, 80, 1, 408, 1, 408, 1, 80, 1, 1, 1, 6, 32, 25, 81, 64, 64, 81, 25, 32, 6, 1

Offset: 0

Andrew Howroyd, Jan 17 2022

The maximum size of an independent set is the independence number which in the case of an m X n king graph is given by ceiling(m/2)*ceiling(n/2).

			Table begins:
=============================================
m\n | 0 1  2  3    4   5     6   7      8
----+----------------------------------------
  0 | 1 1  1  1    1   1     1   1      1 ...
  1 | 1 1  2  1    3   1     4   1      5 ...
  2 | 1 2  4  4   12   8    32  16     80 ...
  3 | 1 1  4  1    9   1    16   1     25 ...
  4 | 1 3 12  9   79  27   408  81   1847 ...
  5 | 1 1  8  1   27   1    64   1    125 ...
  6 | 1 4 32 16  408  64  3600 256  26040 ...
  7 | 1 1 16  1   81   1   256   1    625 ...
  8 | 1 5 80 25 1847 125 26040 625 281571 ...
  ...

Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Cf. A018807, A245013, A332347, A350815, A350819.

T(m,n) = T(n,m).
T(2*m+1, 2*n+1) = 1.
T(2*m, 2*n+1) = (1+m)^(1+n).
T(2*m, 2*n) = A350819(m, n).

cp's OEIS Frontend

A195650 Number of ways to place 11n nonattacking kings on a 22 x 2n chessboard.

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A195651 Number of ways to place 12n nonattacking kings on a 24 x 2n chessboard.

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A319096 Number of nonequivalent ways to place n^2 nonattacking kings on a 2n X 2n chessboard under all symmetry operations of the square.

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A350818 Array read by antidiagonals: T(m,n) is the number of maximum independent sets in the m X n king graph.

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