A236679 -id:A236679 - cp's OEIS frontend

A279113 Number of non-equivalent ways to place 4 non-attacking kings on an n X n board.

0, 0, 1, 14, 277, 2154, 10855, 39926, 120961, 315150, 737089, 1577406, 3150841, 5934034, 10651567, 18332614, 30452605, 49011606, 76753681, 117268590, 175315789, 256949306, 369978631, 524114454, 731604457, 1007394974, 1369985905, 1841600286, 2449309201, 3225197730

Offset: 1

Heinrich Ludwig, Dec 07 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A061997.

			There is 1 way to place 4 non-attacking kings on a 3 X 3 board:
   K.K
   ...
   K.K

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1)

Cf. A061997, A279111 (2 kings), A279112 (3 kings), A279114 (5 kings), A279115 (6 kings), A279116 (7 kings), A279117, A236679.

Table[Boole[n > 2] (n^8 - 54 n^6 + 72 n^5 + 1024 n^4 - 2640 n^3 - 4928 n^2 + 21888 n - 17280 + Boole[OddQ@ n] (14 n^4 - 72 n^3 + 154 n^2 - 240 n - 51))/192, {n, 30}] (* or *)
Rest@ CoefficientList[Series[x^3*(1 + 10 x + 222 x^2 + 1076 x^3 + 2721 x^4 + 2806 x^5 + 1078 x^6 - 924 x^7 - 639 x^8 + 202 x^9 + 236 x^10 - 40 x^11 - 35 x^12 + 6 x^13)/((1 - x)^9*(1 + x)^5), {x, 0, 30}], x] (* Michael De Vlieger, Dec 08 2016 *)

concat(vector(2), Vec(x^3*(1 +10*x +222*x^2 +1076*x^3 +2721*x^4 +2806*x^5 +1078*x^6 -924*x^7 -639*x^8 +202*x^9 +236*x^10 -40*x^11 -35*x^12 +6*x^13) / ((1 -x)^9*(1 +x)^5) + O(x^40))) \\ Colin Barker, Dec 08 2016

a(n) = (n^8 - 54*n^6 + 72*n^5 + 1024*n^4 - 2640*n^3 - 4928*n^2 + 21888*n - 17280 + IF(MOD(n, 2) = 1, 14*n^4 - 72*n^3 + 154*n^2 - 240*n - 51))/192 for n>=3.
a(n) = 4*a(n-1) - a(n-2) - 16*a(n-3) + 19*a(n-4) + 20*a(n-5) - 45*a(n-6) + 45*a(n-8) - 20*a(n-9) - 19*a(n-10) + 16*a(n-11) + a(n-12) - 4*a(n-13) + a(n-14) for n>=17.
G.f.: x^3*(1 +10*x +222*x^2 +1076*x^3 +2721*x^4 +2806*x^5 +1078*x^6 -924*x^7 -639*x^8 +202*x^9 +236*x^10 -40*x^11 -35*x^12 +6*x^13) / ((1 -x)^9*(1 +x)^5). - Colin Barker, Dec 08 2016

A279114 Number of non-equivalent ways to place 5 non-attacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 273, 5335, 50021, 291171, 1263125, 4434783, 13355477, 35672426, 86686721, 194886975, 410820269, 819819261, 1561128613, 2853802623, 5033838173, 8602315716, 14291999441, 23150803815, 36654054741, 56841404455, 86496828245, 129363299967, 190419751685, 276205278030

Offset: 1

Heinrich Ludwig, Dec 08 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A061998.

			There are 273 non-equivalent ways to place 5 non-attacking kings on a 5 X 5 board, e.g., this one:
   K...K
   .....
   ..K..
   .....
   K...K

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A061998, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279115 (6 kings), A279116 (7 kings), A279117, A236679.

[0,0,0] cat [(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960 : n in [4..30]]; // Wesley Ivan Hurt, Dec 08 2016

A279114:=n->(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960: 0, 0, 0, seq(A279114(n), n=4..30); # Wesley Ivan Hurt, Dec 08 2016

Join[{0, 0, 0}, Table[(n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2 - (-1)^n/2)*(52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960, {n, 4, 30}]] (* Wesley Ivan Hurt, Dec 08 2016 *)

concat(vector(4), Vec(x^5*(273 +3970*x +24438*x^2 +67866*x^3 +103134*x^4 +66494*x^5 -1418*x^6 -29015*x^7 -4247*x^8 +10650*x^9 +2718*x^10 -2696*x^11 -672*x^12 +382*x^13 +62*x^14 -19*x^15) / ((1 -x)^11*(1 +x)^6) + O(x^30))) \\ Colin Barker, Dec 08 2016

a(n) = (n^10 - 90*n^8 + 120*n^7 + 3115*n^6 - 7748*n^5 - 46050*n^4 + 173140*n^3 + 158584*n^2 - 1255952*n + 1252800 + (1/2-(-1)^n/2) * (52*n^5 - 305*n^4 + 180*n^3 + 320*n^2 + 3488*n - 375))/960 for n >= 4.
a(n) = 5*a(n-1) - 4*a(n-2) - 20*a(n-3) + 40*a(n-4) + 16*a(n-5) - 100*a(n-6) + 44*a(n-7) + 110*a(n-8) - 110*a(n-9) - 44*a(n-10) + 100*a(n-11) - 16*a(n-12) - 40*a(n-13) + 20*a(n-14) + 4*a(n-15) - 5*a(n-16) + a(n-17) for n >= 21.
G.f.: x^5*(273 +3970*x +24438*x^2 +67866*x^3 +103134*x^4 +66494*x^5 -1418*x^6 -29015*x^7 -4247*x^8 +10650*x^9 +2718*x^10 -2696*x^11 -672*x^12 +382*x^13 +62*x^14 -19*x^15) / ((1 -x)^11*(1 +x)^6). - Colin Barker, Dec 08 2016

A279115 Number of non-equivalent ways to place 6 non-attacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 143, 7855, 153311, 1505465, 9729830, 47235703, 186615092, 630338668, 1882894541, 5092130575, 12686490993, 29498296651, 64664954532, 134715649055, 268438970166, 514318521438, 951646716171, 1706721390223, 2976056379875, 5058962536429, 8402677784738, 13663807273607

Offset: 1

Heinrich Ludwig, Dec 09 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A172158.

			There are 143 non-equivalent ways to place 6 non-attacking kings on a 5 X 5 board, e.g., this one:
   K...K
   .....
   K...K
   .....
   K...K

Heinrich Ludwig, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8,-22,69,-8,-176,168,182,-364,0,364,-182,-168,176,8,-69,22,8,-6,1).

Cf. A172158, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279114 (5 kings), A279116 (7 kings), A279117, A236679.

concat(vector(4), Vec(x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7) + O(x^30))) \\ Colin Barker, Dec 09 2016

a(n) = (n^12 - 135*n^10 + 180*n^9 + 7465*n^8 - 18840*n^7 - 202468*n^6 + 749880*n^5 + 2446764*n^4 - 13439400*n^3 - 3570352*n^2 + 89413920*n - 107694720 + IF(MOD(n, 2) = 1, 122*n^6 - 1020*n^5 + 1955*n^4 + 840*n^3 + 5753*n^2 - 42840*n + 132975))/5760 for n>=5.
a(n) = 6*a(n-1) - 8*a(n-2) - 22*a(n-3) + 69*a(n-4) - 8*a(n-5) - 176*a(n-6) + 168*a(n-7) + 182*a(n-8) - 364*a(n-9) + 364*a(n-11) - 182*a(n-12) - 168*a(n-13) + 176*a(n-14) + 8*a(n-15) - 69*a(n-16) + 22*a(n-17) + 8*a(n-18) - 6*a(n-19) + a(n-20) for n>=25.
G.f.: x^5*(143 +6997*x +107325*x^2 +651585*x^3 +2086471*x^4 +3732434*x^5 +3669293*x^6 +1297859*x^7 -708745*x^8 -592136*x^9 +247421*x^10 +258649*x^11 -53671*x^12 -77714*x^13 +4451*x^14 +14969*x^15 +1018*x^16 -1741*x^17 -234*x^18 +106*x^19) / ((1 -x)^13*(1 +x)^7). - Colin Barker, Dec 09 2016

A279116 Number of non-equivalent ways to place 7 non-attacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 39, 6472, 311552, 5576682, 56289710, 389130774, 2059061646, 8924241327, 33134160010, 108698226956, 322211640480

Offset: 1

Heinrich Ludwig, Dec 10 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A194788.

			There are 39 non-equivalent ways to place 7 non-attacking kings on a 5 X 5 board, e.g., this one:
   K...K
   .....
   K.K.K
   .....
   K...K

Cf. A194788, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279114 (5 kings), A279115 (6 kings), A279117, A236679.

A279117 Number of non-equivalent ways to place n non-attacking kings on an n X n board.

Original entry on oeis.org

1, 0, 2, 14, 273, 7855, 311552, 14895797, 831959075, 52959962415

Offset: 1

Heinrich Ludwig, Dec 10 2016

Rotations and reflections of placements are not counted. If they are to be counted, see A201513.

			There are 14 non-equivalent ways to place 4 non-attacking kings on a 4 X 4 board, e.g., this one:
   K..K
   ....
   ....
   K..K

Cf. A201513, A279111 (2 kings), A279112 (3 kings), A279113 (4 kings), A279114 (5 kings), A279115 (6 kings), A279116 (7 kings), A236679.

A286443 Irregular triangle read by rows: T(n, k) = number of non-equivalent ways to tile an n X n X n triangular area with k 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-4*k) of 1 X 1 X 1 tiles.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 4, 10, 14, 6, 1, 6, 32, 97, 142, 105, 46, 14, 3, 1, 1, 8, 70, 398, 1280, 2386, 2574, 1569, 524, 87, 3, 1, 11, 143, 1290, 7301, 26471, 62067, 94423, 93358, 60287, 25881, 7697, 1678, 281, 40, 5, 1, 1, 13, 252, 3366, 29603, 176591, 728868

Offset: 1

Heinrich Ludwig, May 16 2017

The triangle T(n, k) is irregularly shaped: For n >= 4: 0 <= k <= n^2/4 if n is even, 0 <= k <= (n^2 -9)/4 if n is odd. First row corresponds to n = 1.
Rotations and reflections of tilings are not counted. If they are to be counted, see A286436. Tiles of the same size are indistinguishable.
For an analogous problem concerning square tiles, see A236679.

			The triangle begins with T(1, 0)
   1;
   1,    1;
   1,    1;
   1,    3,    3,    2,    1;
   1,    4,   10,   14,    6;
   1,    6,   32,   97,  142,  105,   46,   14,    3,    1;
   1,    8,   70,  398, 1280, 2386, 2574, 1569,  524,   87,    3;
T(4, 3) = 2 because there are 2 non-equivalent ways to tile an area of size 4X4X4 with 3 tiles of size 2X2X2 and fill up the rest with tiles of size 1X1X1.

Heinrich Ludwig, Table of n, a(n) for n = 1..140

Cf. A236679, A286436, A286444, A286445, A286446.

A275869 Number of nonequivalent ways to place k>=0 nonattacking kings on an n X n board.

Original entry on oeis.org

2, 2, 11, 51, 922, 25618, 1597350, 169413040, 33716225195, 11838480390673, 7588965091313449, 8705702554970941096, 18079208010076976255573, 67519585404524909086260614, 455193583190737164106702088892, 5527752160260327852724215089473548

Offset: 1

Heinrich Ludwig, Dec 11 2016

nonn

Also number of nonequivalent ways to tile an n+1 X n+1 square with 1 X 1 and 2 X 2 tiles.
Also row sum of triangle A236679.
Rotations and reflections of a placement are not counted. If they are to be counted, see A063443.

Andrew Howroyd, Table of n, a(n) for n = 1..30

Cf. A236679, A063443.

a(10)-a(16) from Andrew Howroyd, May 30 2017

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

A362259 Maximum number of ways in which a set of integer-sided squares can tile an n X n square, up to rotations and reflections.

Original entry on oeis.org

1, 1, 1, 1, 4, 20, 277, 7855, 487662

Offset: 0

Pontus von Brömssen, Apr 15 2023

Main diagonal of A362258.

Cf. A224239, A236679, A361222 (rectangular pieces), A362143.

a(n) >= A362143(n)/8.

cp's OEIS Frontend

A279113 Number of non-equivalent ways to place 4 non-attacking kings on an n X n board.

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A279114 Number of non-equivalent ways to place 5 non-attacking kings on an n X n board.

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A279115 Number of non-equivalent ways to place 6 non-attacking kings on an n X n board.

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A279116 Number of non-equivalent ways to place 7 non-attacking kings on an n X n board.

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A279117 Number of non-equivalent ways to place n non-attacking kings on an n X n board.

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A286443 Irregular triangle read by rows: T(n, k) = number of non-equivalent ways to tile an n X n X n triangular area with k 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-4*k) of 1 X 1 X 1 tiles.

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A275869 Number of nonequivalent ways to place k>=0 nonattacking kings on an n X n board.

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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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A362259 Maximum number of ways in which a set of integer-sided squares can tile an n X n square, up to rotations and reflections.

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