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.

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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.

Original entry on oeis.org

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

Views

Author

Anton Nikonov, Dec 21 2018

Keywords

Comments

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

Examples

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

Crossrefs

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

Formula

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

Extensions

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

A321614 Number of nonequivalent ways to place 2n nonattacking kings on a 4 X 2n chessboard under all symmetry operations of the rectangle.

Original entry on oeis.org

1, 4, 23, 106, 473, 1939, 7618, 28703, 105112, 375597, 1316944, 4544124, 15474559, 52108212, 173799309, 574908646, 1888125243, 6162032375, 19998659760, 64584817367, 207655073310, 665017743665
Offset: 0

Views

Author

Anton Nikonov, Dec 19 2018

Keywords

Comments

A maximum of 2n nonattacking kings can be placed on a 4 X 2n chessboard.
Number of nonequivalent ways of placing 2n 2 X 2 tiles in an 5 X (2n+1) rectangle under all symmetry operations of the rectangle. - Andrew Howroyd, Dec 21 2018

Crossrefs

Cf. A001787, A061593, A061594, A231145, A319096, A322284.

Formula

a(n) = A231145(2*n+1, 2n).
Conjectures from Colin Barker, Dec 22 2018: (Start)
G.f.: (1 - 2*x)*(1 - 6*x + 17*x^2 - 18*x^3 - 2*x^4 + 7*x^5 + 6*x^6 - 3*x^7) / ((1 - x)^2*(1 - 3*x)^2*(1 - 3*x + x^2)*(1 - x - x^2)*(1 - 3*x^2)).
a(n) = 12*a(n-1) - 54*a(n-2) + 98*a(n-3) + 17*a(n-4) - 346*a(n-5) + 505*a(n-6) - 210*a(n-7) - 120*a(n-8) + 126*a(n-9) - 27*a(n-10) for n>9.
(End)
Showing 1-2 of 2 results.

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