A061594 -id:A061594 - 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).

A061593 Number of ways to place 2n nonattacking kings on a 4 X 2n chessboard.

Original entry on oeis.org

12, 79, 408, 1847, 7698, 30319, 114606, 419933, 1501674, 5266069, 18174084, 61892669, 208424880, 695179339, 2299608732, 7552444115, 24648046806, 79994460139, 258339007890, 830619734681, 2660070154542, 8488515938929, 27000079296648, 85629004867577

Offset: 1

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001

Bruno Berselli, Table of n, a(n) for n = 1..200
D. E. Knuth, Nonattacking kings on a chessboard, 1994.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 90.
H. S. Wilf, The problem of the kings, Elec. J. Combin. 2, 1995.
Index entries for linear recurrences with constant coefficients, signature (9,-28,33,-9).

Column k=2 of A350819.

Cf. A061594, A001787.

[(17*n-109)*3^n+2*Fibonacci(2*n+10): n in [1..30]]; // Vincenzo Librandi, Jul 12 2011

with(combinat): A061593:=n->(17*n-109)*3^n+2*fibonacci(2*n+10): seq(A061593(n), n=1..30); # Wesley Ivan Hurt, Nov 08 2014

Table[(17 n - 109)*3^n + 2 Fibonacci[2 n + 10], {n, 30}] (* Wesley Ivan Hurt, Nov 08 2014 *)
CoefficientList[Series[x (12-29x+33x^2-9x^3)/((1-3x+x^2)(1-3x)^2),{x,0,30}],x] (* or *) LinearRecurrence[{9,-28,33,-9},{0,12,79,408,1847},30] (* Harvey P. Dale, Dec 20 2021 *)

G.f.: x*(12-29*x+33*x^2-9*x^3)/((1-3*x+x^2)*(1-3*x)^2).
a(n) = 9*a(n-1) - 28*a(n-2) + 33*a(n-3) - 9*a(n-4); a(1)=12, a(2)=79, a(3)=408, a(4)=1847.
a(n) = (17*n-109)*3^n + 2*Fibonacci(2*n+10).
a(n) = 17*A027471(n+2) - 126*A000244(n) + A025169(n+4).

A173782 Number of ways to place 4n nonattacking kings on an 8 X 2n chessboard.

Original entry on oeis.org

80, 1847, 26040, 281571, 2580754, 21137959, 159636030, 1134127305, 7683664202, 50123713793, 317076250136, 1955475353217, 11806000507544, 70004699407151, 408747986045656, 2355077855615435, 13413115039118042, 75623103424916527

Offset: 1

Vaclav Kotesovec, Feb 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (44, -887, 10855, -90083, 536398, -2365292, 7860674, -19852652, 38152568, -55523880, 60518766, -48502595, 27783210, -10888525, 2721025, -382125, 22500).

Column k=4 of A350819.

Cf. A061594, A061593, A018807.

CoefficientList[Series[(22500 x^16 - 382125 x^15 + 2723005 x^14 - 10917322 x^13 + 27938661 x^12 - 48873227 x^11 + 60780149 x^10 - 54895129 x^9 + 36368733 x^8 - 17776175 x^7 + 6499001 x^6 - 1854479 x^5 + 446565 x^4 - 94300 x^3 + 15732 x^2 - 1673 x + 80) / ((1 - x) (x^2 - 4 x + 1) (x^3 - 6 x^2 + 5 x - 1) (4 x - 1) (5 x - 1)^2 (3 x^2 - 5 x + 1)^2 (5 x^2 - 5 x + 1)^2),{x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

G.f.: x*(22500*x^16 -382125*x^15 +2723005*x^14 -10917322*x^13 +27938661*x^12 -48873227*x^11 +60780149*x^10 -54895129*x^9 +36368733*x^8 -17776175*x^7 +6499001*x^6 -1854479*x^5+446565*x^4 -94300*x^3 +15732*x^2 -1673*x+80) / ((1-x) *(x^2-4*x+1) *(x^3-6*x^2+5*x-1) *(4*x-1) *(5*x-1)^2 *(3*x^2-5*x+1)^2 *(5*x^2-5*x+1)^2).
Recurrence: a(n) = 44a(n-1) -887a(n-2) +10855a(n-3) -90083a(n-4) +536398a(n-5) -2365292a(n-6) +7860674a(n-7) -19852652a(n-8) +38152568a(n-9) -55523880a(n-10) +60518766a(n-11) -48502595a(n-12) +27783210a(n-13) -10888525a(n-14) +2721025a(n-15) -382125a(n-16) +22500a(n-17), n>17.
a(n) = (-12505804889/302760 +7963567/2610*n)*5^n +3872/3*4^n -1/24 +(135343*sqrt(3)/18 -234421/18)*(2 -sqrt(3))^n -(135343*sqrt(3)/18 +234421/18)*(2 +sqrt(3))^n +(33301/5 -74461*sqrt(5)/25 +(141*sqrt(5)/25 -63/5)*n)*((5 -sqrt(5))/2)^n +(74461*sqrt(5)/25 +33301/5 - (141*sqrt(5)/25 + 63/5)*n)*((5 +sqrt(5))/2)^n + (4306740/169 - 1194474*sqrt(13)/169 + (139103/117 - 501541*sqrt(13)/1521)*n)*((5 -sqrt(13))/2)^n +(1194474*sqrt(13)/169 +4306740/169 +(501541*sqrt(13)/1521 +139103/117)*n)*((5 +sqrt(13))/2)^n +72*(b*(3504697*c - 11380560) -11380560*c +36953816)/(142129*(a - b)*(a - c))*a^n +72*(a*(3504697*c - 11380560) - 8*(1422570*c - 4619227))/(142129*(a - b)*(c - b))*b^n +72*(a*(3504697*b - 11380560) -8*(1422570*b - 4619227))/(142129*(a - c)*(b - c))*c^n, where: a=2-2*sin(Pi/14), b=2+2*sin(3*Pi/14), c=2-2*cos(Pi/7). - Vaclav Kotesovec, added Mar 01 2010, updated Mar 29 2010.

A173783 Number of ways to place 5n nonattacking kings on a 10 X 2n chessboard.

Original entry on oeis.org

192, 7698, 166368, 2580754, 32572756, 357365350, 3544192112, 32580145116, 282359109140, 2335042206624, 18589546217696, 143422674213726, 1077891352444220, 7923134615854816, 57146364209686016, 405497952834408698

Offset: 1

Vaclav Kotesovec, Feb 24 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Index entries for linear recurrences with constant coefficients, signature (85, -3441, 88303, -1613002, 22327010, -243429637, 2145452227, -15565947848, 94202823084, -480152808502, 2075863416838, -7651361422835, 24128330540449, -65240466585284, 151411770874148, -301613628545814, 515173613407544, -753006145475828, 939001403456656, -994821988961592, 890558910282768, -668920434927504, 417832289937792, -214574645977920, 89258591798784, -29486236792320, 7526493775872, -1426182018048, 188221833216, -15390756864, 585252864).

Column k=5 of A350819.

Cf. A173782, A061594, A061593, A018807.

CoefficientList[Series[2 (292626432 x^30 - 7695378432 x^29 + 94084706304 x^28 - 712519981056 x^27 + 3757888797696 x^26 - 14715718076160 x^25 + 44556058968960 x^24 - 107273952716256 x^23 + 209645023363168 x^22 - 337824014576768 x^21 + 454329405135504 x^20 - 514643686425920 x^19 + 494203416082160 x^18 - 403847150294172 x^17 + 281135354205764 x^16 - 166453721883480 x^15 + 83456844800670 x^14 - 35182845104124 x^13 + 12345883162136 x^12 - 3557728594620 x^11 + 827346101101 x^10 - 152042822189 x^9 + 21726065190 x^8 - 2499103126 x^7 + 289877178 x^6 - 45817212 x^5 + 7810422 x^4 - 1012942 x^3 + 86355 x^2 - 4311 x + 96) / ((1 - 2 x) (x^2 - 4 x + 1) (4 x - 1) (6 x - 1)^2 (2 x^2 - 4 x + 1) (2 x^2 - 5 x + 1) (4 x^2 - 6 x + 1)^2 (6 x^2 - 6 x + 1)^2 (7 x^2 - 6 x + 1)^2 (2 x^3 - 8 x^2 + 6 x - 1) (3 x^3 - 9 x^2 + 6 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

G.f.: 2*x*(292626432*x^30 -7695378432*x^29 +94084706304*x^28 -712519981056*x^27 +3757888797696*x^26 -14715718076160*x^25 +44556058968960*x^24 -107273952716256*x^23 +209645023363168*x^22 -337824014576768*x^21 +454329405135504*x^20 -514643686425920*x^19 +494203416082160*x^18 -403847150294172*x^17 +281135354205764*x^16 -166453721883480*x^15 +83456844800670*x^14 -35182845104124*x^13 +12345883162136*x^12 -3557728594620*x^11 +827346101101*x^10 -152042822189*x^9 +21726065190*x^8 -2499103126*x^7 +289877178*x^6 -45817212*x^5 +7810422*x^4 -1012942*x^3 +86355*x^2 -4311*x+96) / ((1-2*x) *(x^2-4*x+1) *(4*x-1) *(6*x-1)^2 *(2*x^2-4*x+1) *(2*x^2-5*x+1) *(4*x^2-6*x+1)^2 *(6*x^2-6*x+1)^2 *(7*x^2-6*x+1)^2 *(2*x^3-8*x^2+6*x-1) *(3*x^3-9*x^2+6*x-1)^2).

Recurrence: a(n) = 85a(n-1) -3441a(n-2) +88303a(n-3) -1613002a(n-4) +22327010a(n-5) -243429637a(n-6) +2145452227a(n-7) -15565947848a(n-8) +94202823084a(n-9) -480152808502a(n-10) +2075863416838a(n-11) -7651361422835a(n-12) +24128330540449a(n-13) -65240466585284a(n-14) +151411770874148a(n-15) -301613628545814a(n-16) +515173613407544a(n-17) -753006145475828a(n-18) +939001403456656a(n-19) -994821988961592a(n-20) +890558910282768a(n-21) -668920434927504a(n-22) +417832289937792a(n-23) -214574645977920a(n-24) +89258591798784a(n-25) -29486236792320a(n-26) +7526493775872a(n-27) -1426182018048a(n-28) +188221833216a(n-29) -15390756864a(n-30) +585252864a(n-31), n>31.

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

A194645 Number of ways to place 3n nonattacking kings on a 6 X 2n cylindrical chessboard.

Original entry on oeis.org

32, 100, 344, 1220, 4392, 15988, 58776, 218052, 815816, 3076180, 11682296, 44653028, 171670440, 663421684, 2575592664, 10039703172, 39273896840, 154109956756, 606353229752, 2391296071460, 9449664931176, 37407140524084, 148300497571992, 588693691298244

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 = 6, number of rows = 2n).

Ray Chandler, Table of n, a(n) for n = 1..1660
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (12,-53,104,-86,24).

Cf. A061594, A194644, A137432, A195591.

Table[FullSimplify[2*4^n+2*3^n+4*(2+Sqrt[2])^n+4*(2-Sqrt[2])^n+2], {n,25}]
LinearRecurrence[{12,-53,104,-86,24},{32,100,344,1220,4392},30] (* Harvey P. Dale, Jul 25 2016 *)

a(n) = 2*4^n + 2*3^n + 4*(2+sqrt(2))^n + 4*(2-sqrt(2))^n + 2.
Recurrence: a(n) = 24*a(n-5) - 86*a(n-4) + 104*a(n-3) - 53*a(n-2) + 12*a(n-1).
G.f.: -2*(7-68*x+229*x^2-308*x^3+134*x^4)/((-1+x)*(-1+3*x)*(-1+4*x)*(1-4*x+2*x^2)).

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

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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A061593 Number of ways to place 2n nonattacking kings on a 4 X 2n chessboard.

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A173782 Number of ways to place 4n nonattacking kings on an 8 X 2n chessboard.

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A173783 Number of ways to place 5n nonattacking kings on a 10 X 2n chessboard.

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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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A194645 Number of ways to place 3n nonattacking kings on a 6 X 2n cylindrical chessboard.

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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.