A173783 -id:A173783 - cp's OEIS frontend

A018807 Number of ways to place n^2 nonattacking kings on 2n X 2n chessboard.

1, 4, 79, 3600, 281571, 32572756, 5109144543, 1027533353168, 254977173389319, 75925129079783308, 26568150968269086211, 10749154284380665611224, 4963704194366362387891227, 2588716234142991968960920692, 1511548995678989691821551648635

Offset: 0

David W. Wilson

Rotations and reflections are considered distinct.
Also, number of ways to tile a (2n+1) X (2n+1) board with n^2 2 X 2 tiles and 4n+1 1 X 1 tiles, rotations and reflections counted as distinct. - David W. Wilson, Aug 18 2011
Number of maximum independent vertex sets in the 2n X 2n king graph. - Eric W. Weisstein, Jun 20 2017

David W. Wilson, Table of n, a(n) for n = 0..26
Zealint Blog (Russian) Source for a(12) through a(20), March 14 2011. a(21) through a(26) from same source, July 9 2011.
Tricia Muldoon Brown, Queens, Attack!, The Mathematical Intelligencer (2020).
Tricia Muldoon Brown, Maximum arrangements of nonattacking kings on the 2n × 2n chessboard, Georgia Southern University (2020).
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 160-162.
Michael Larsen, The Problem of Kings, The Electronic Journal of Combinatorics 2, 1995
Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set

Main diagonal of A350819.

Cf. A174558, A174155, A174154, A173782, A173783, A061594, A061593.

Asymptotic (M. Larsen, 1995): log(a(n)) = 2n*log(n) - 2n*log(2) + O(n^(4/5)*log(n)).

a(0) added by Geoffrey H. Morley, Feb 06 2013

A350819 Array read by antidiagonals: T(m,n) is the number of maximum independent sets in the 2m X 2n king graph.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 79, 32, 1, 1, 80, 408, 408, 80, 1, 1, 192, 1847, 3600, 1847, 192, 1, 1, 448, 7698, 26040, 26040, 7698, 448, 1, 1, 1024, 30319, 166368, 281571, 166368, 30319, 1024, 1, 1, 2304, 114606, 976640, 2580754, 2580754, 976640, 114606, 2304, 1

Offset: 0

Andrew Howroyd, Jan 17 2022

Number of ways to tile a (2m+1) X (2n+1) board with m*n 2 X 2 tiles and 2m+2n+1 1 X 1 tiles.

For m,n > 0, T(m,n) is the number of minimum dominating sets in the (3m-1) X (3n-1) king graph.

			Table begins:
=============================================
m\n | 0   1    2      3       4        5
----+----------------------------------------
  0 | 1   1    1      1       1        1 ...
  1 | 1   4   12     32      80      192 ...
  2 | 1  12   79    408    1847     7698 ...
  3 | 1  32  408   3600   26040   166368 ...
  4 | 1  80 1847  26040  281571  2580754 ...
  5 | 1 192 7698 166368 2580754 32572756 ...
  ...

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

Rows 0..12 are A000012, A001787(n+1), A061593, A061594, A173782, A173783, A174154, A174155, A174558, A195648, A195649, A195650, A195651.
Main diagonal is A018807.
Cf. A350815, A350818.

T(m,n) = T(n,m).

T(m,n) = A350818(2*m, 2*n) = A350815(3*m-1, 3*n-1).

A174154 Number of ways to place 6n nonattacking kings on a 12 x 2n chessboard.

Original entry on oeis.org

1, 448, 30319, 976640, 21137959, 357365350, 5109144543, 64737165162, 749160010737, 8080813574550, 82425144219429, 803491953235264, 7545414941610145, 68680800264413920, 608889093898882615, 5278006575696293456, 44873569636443901967, 375159494582050088590, 3090799708762482416287

Offset: 0

Vaclav Kotesovec, Mar 10 2010

nonn

Ray Chandler, Table of n, a(n) for n = 0..1172
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Vaclav Kotesovec, G.f.
Index entries for linear recurrences with constant coefficients, order 75.

Column k=6 of A350819.

Cf. A061594, A061593, A018807, A173782, A173783.

More terms from Jinyuan Wang, Feb 26 2020

a(0)=1 prepended by Andrew Howroyd, Mar 26 2023

A174558 Number of ways to place 8n nonattacking kings on a 16 x 2n chessboard.

Original entry on oeis.org

2304, 419933, 28432288, 1134127305, 32580145116, 749160010737, 14677177838054, 254977173389319, 4035559337688370, 59315924213143597, 821112680030028632, 10819171744710664383, 136800806311499633208, 1670597119210336446533, 19804685547188544317522, 228865023358344707514899, 2586924156960003793687130, 28681715460054576813151389, 312656761422008821513384848, 3357651442822195404605813501

Offset: 1

Vaclav Kotesovec, Nov 29 2010

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013
Vaclav Kotesovec, G.f.
Index entries for linear recurrences with constant coefficients, order 307.

Column k=8 of A350819.

Cf. A174155, A174154, A173782, A173783, A061594, A061593, A018807.

Asymptotic formula for number of ways to place m x n nonattacking kings on a 2m x 2n chessboard (this case is m=8): f(m,n) ~ k(m)*n*(m+1)^n
First values of k(m):
k(1)=1,
k(2)=17,
k(3)=231,
k(4)=3051.17509,
k(5)=40881.99638,
k(6)=563050.92363,
k(7)=8008508.28858,
k(8)=117833087.45133
k(9)=1794306724.77472
k(10)=28276454469.76459
k(11)=461049875818.05305
k(12)=7775513990776.97046
k(13)=135589372611110.17367
k(14)=2443990803097108.58764
k(15)=45522076785406201.22572
k(16)=875939597341977670.66777
k(17)=17407856624734801679.11613
k(18)=357216046100723515478.42809
k(19)=7567101689641721175327.80272

A174155 Number of ways to place 7n nonattacking kings on a 14 x 2n chessboard.

Original entry on oeis.org

1, 1024, 114606, 5392704, 159636030, 3544192112, 64737165162, 1027533353168, 14677177838054, 193194265398240, 2383116363555182, 27889602664055396, 312546900470579954, 3378090945290324892, 35412239480510055916, 361670315347336810428, 3611858972942315054336, 35375586671457852212944

Offset: 0

Vaclav Kotesovec, Mar 10 2010

nonn

Ray Chandler, Table of n, a(n) for n = 0..1096
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Vaclav Kotesovec, G.f.
Index entries for linear recurrences with constant coefficients, order 124.

Column k=7 of A350819.

Cf. A018807, A061594, A061593, A173782, A173783, A174154.

More terms from Jinyuan Wang, Feb 26 2020

a(0)=1 prepended by Andrew Howroyd, Mar 26 2023

A194647 Number of ways to place 5n nonattacking kings on a 10 X 2n cylindrical chessboard.

Original entry on oeis.org

192, 708, 3036, 13932, 66532, 327192, 1649420, 8500668, 44693472, 239238888, 1301236304, 7177627944, 40078823652, 226167613792, 1287874058656, 7390391650172, 42688584938548, 247956702607932, 1447080255512308, 8479116559291112, 49852445684576540

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 10, number of rows = 2n).

Ray Chandler, Table of n, a(n) for n = 1..1284
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Vaclav Kotesovec, Explicit formula and recurrence
Index entries for linear recurrences with constant coefficients, signature (56, -1460, 23541, -263028, 2162701, -13565686, 66416673, -257594833, 798883747, -1991743054, 4000492482, -6469161690, 8395031504, -8690399936, 7111450512, -4541105512, 2222092032, -811893408, 213097152, -37748736, 4020480, -193536).

Cf. A173783, A194644, A194645, A194646, A137432, A195593.

G.f.: -2*(7089408*x^21 - 132938496*x^20 + 1125112128*x^19 - 5717239392*x^18 + 19578445344*x^17 - 48082847384*x^16 + 88003026752*x^15 - 123138008952*x^14 + 134072006560*x^13 - 114991853490*x^12 + 78336556962*x^11 - 42596878318*x^10 + 18524447581*x^9 - 6435525481*x^8 + 1778018953*x^7 - 387290192*x^6 + 65568715*x^5 - 8436954*x^4 + 796245*x^3 - 51918*x^2 + 2088*x - 39)/((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(x^2-4*x+1)*(2*x^2-5*x+1)*(2*x^2-4*x+1)*(4*x^2-6*x+1)*(6*x^2-6*x+1)*(7*x^2-6*x+1)*(2*x^3-8*x^2+6*x-1)*(3*x^3-9*x^2+6*x-1)).

Asymptotic: a(n) ~ 2*6^n.

A195593 Number of ways to place 5n nonattacking kings on a vertical cylinder 10 X 2n.

Original entry on oeis.org

64, 732, 4392, 18890, 66532, 205628, 580664, 1536814, 3877300, 9434784, 22327496, 51698178, 117645348, 263992580, 585640568, 1286898262, 2805399156, 6074441896, 13076687560, 28009586346, 59732295204, 126891641612, 268638308152, 566987715710

Offset: 1

Vaclav Kotesovec, Sep 21 2011

nonn

Vertical cylinder: a chessboard where it is supposed that the columns 1 and 10 are in contact (number of columns = 10, number of rows = 2n).

Ray Chandler, Table of n, a(n) for n = 1..3299 (first 1000 terms from Jinyuan Wang)
Vaclav Kotesovec, Non-attacking chess pieces
Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Cf. A194647, A173783, A137432.

LinearRecurrence[{8, -26, 44, -41, 20, -4}, {64, 732, 4392, 18890, 66532, 205628}, 20] (* Jinyuan Wang, Feb 26 2020 *)

a(n) = -4*a(n-6) + 20*a(n-5) - 41*a(n-4) + 44*a(n-3) - 26*a(n-2) + 8*a(n-1).
G.f.: (1 + 56*x + 246*x^2 + 156*x^3 + 11*x^4)/((x-1)^4*(2*x-1)^2).
a(n) = (1771*n - 8709)*2^n + 235/3*n^3 + 880*n^2 + 12815/3*n + 8710.

cp's OEIS Frontend

A018807 Number of ways to place n^2 nonattacking kings on 2n X 2n chessboard.

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A350819 Array read by antidiagonals: T(m,n) is the number of maximum independent sets in the 2m X 2n king graph.

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A174154 Number of ways to place 6n nonattacking kings on a 12 x 2n chessboard.

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A174558 Number of ways to place 8n nonattacking kings on a 16 x 2n chessboard.

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A174155 Number of ways to place 7n nonattacking kings on a 14 x 2n chessboard.

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A194647 Number of ways to place 5n nonattacking kings on a 10 X 2n cylindrical chessboard.

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A195593 Number of ways to place 5n nonattacking kings on a vertical cylinder 10 X 2n.

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