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User: Anton Nikonov

Anton Nikonov has authored 3 sequences.

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

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

Offset: 0

Anton Nikonov, Dec 19 2018

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

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

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)

A322284 Number of nonequivalent ways to place n nonattacking kings on a 2 X 2n chessboard under all symmetry operations of the rectangle.

Original entry on oeis.org

1, 4, 8, 22, 48, 116, 256, 584, 1280, 2832, 6144, 13344, 28672, 61504, 131072, 278656, 589824, 1245440, 2621440, 5505536, 11534336, 24118272, 50331648, 104859648, 218103808, 452988928, 939524096, 1946165248, 4026531840, 8321515520, 17179869184, 35433512960

Offset: 1

Anton Nikonov, Dec 02 2018

A maximum of n nonattacking kings can be placed on a 2 X 2n chessboard.
Number of nonequivalent ways of placing n 2 X 2 tiles in an 3 X (2n+1) rectangle under all symmetry operations of the rectangle. - Andrew Howroyd, Dec 16 2018
Number of ways to choose modulo symmetry n vertices from a 1 X (2n-1) square grid with distances > sqrt(2) between the vertices. (Consider the interior 1 X (2*n-1) square grid of the 3 X (2n+1) square grid, or the square grid with the midpoints of the squares of the  2 X 2n chessboard as vertices.) - Wolfdieter Lang, Feb 07 2019

			For n = 2 there are a(2) = 4 distinct solutions from 12 that will not be repeated by all possible turns and reflections.
1.                  2.                 3.                 4.
-----------------   -----------------  -----------------  -----------------
| * |   | * |   |   | * |   |   | * |  | * |   |   |   |  | * |   |   |   |
-----------------   -----------------  -----------------  -----------------
|   |   |   |   |   |   |   |   |   |  |   |   | * |   |  |   |   |   | * |
-----------------   -----------------  -----------------  -----------------

Muniru A Asiru, Table of n, a(n) for n = 1..3280
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).

Cf. A001787, A061593, A061594, A238009, A321614.

seq(coeff(series(x*(1-6*x^2+6*x^3)/((1-2*x)^2*(1-2*x^2)),x,n+1), x, n), n = 1 .. 35); # Muniru A Asiru, Dec 21 2018

Vec(x*(1 - 6*x^2 + 6*x^3) / ((1 - 2*x)^2*(1 - 2*x^2)) + O(x^40)) \\ Colin Barker, Dec 21 2018

a(n) = (n+1)*2^(n-2) + (1 + (-1)^n)^(n/2 - 1) for n > 1.
a(n) = A238009(2*n+1, n). - Andrew Howroyd, Dec 16 2018
From Colin Barker, Dec 21 2018: (Start)
G.f.: x*(1 - 6*x^2 + 6*x^3) / ((1 - 2*x)^2*(1 - 2*x^2)).
a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 8*a(n-4) for n>4. (End)
E.g.f.: (exp(2*x)*(1 + 2*x) + 2*cosh(sqrt(2)*x) - 3)/4. - Stefano Spezia, May 14 2023

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

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User: Anton Nikonov

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

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A322284 Number of nonequivalent ways to place n nonattacking kings on a 2 X 2n chessboard under all symmetry operations of the rectangle.

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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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