A061998 -id:A061998 - cp's OEIS frontend

A061995 Number of ways to place 2 nonattacking kings on an n X n board.

0, 0, 0, 16, 78, 228, 520, 1020, 1806, 2968, 4608, 6840, 9790, 13596, 18408, 24388, 31710, 40560, 51136, 63648, 78318, 95380, 115080, 137676, 163438, 192648, 225600, 262600, 303966, 350028, 401128, 457620, 519870, 588256

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A061996, A061997, A061998.

[0] cat [(n-1)*(n-2)*(n^2+3*n-2)/2: n in [1..30]]; // G. C. Greubel, Nov 04 2018

CoefficientList[Series[2 x^3 (-8 + x + x^2) / (x-1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

x='x+O('x^30); Vec(2*x^3*(x^2+x-8)/(x-1)^5) \\ G. C. Greubel, Nov 04 2018

G.f.: 2*x^3*(x^2 + x - 8)/(x - 1)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), n >= 6.
a(n) = (n - 1)*(n - 2)*(n^2 + 3*n - 2)/2, n >= 1.
E.g.f.: (4 - (4 - 4*x + 2*x^2 - 6*x^3 - x^4)*exp(x))/2. - G. C. Greubel, Nov 04 2018

A061996 Number of ways to place 3 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 8, 140, 964, 3920, 11860, 29708, 65240, 129984, 240240, 418220, 693308, 1103440, 1696604, 2532460, 3684080, 5239808, 7305240, 10005324, 13486580, 17919440, 23500708, 30456140, 39043144, 49553600, 62316800

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Placing non-attacking queens and kings on boards of various sizes, part of "Between chessboard and computer", 1996, pp. 204 - 206.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A061995, A061997, A061998, A193580.

CoefficientList[Series[4x^3(2 +21x +38x^2 -42x^3 +11x^4)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

[(n-1)*(n-2)*(n^4+3*n^3-20*n^2-30*n+132)/6 -44*bool(n==0) for n in (0..40)] # G. C. Greubel, Apr 29 2022

G.f.: 4*x^3*(2 + 21*x + 38*x^2 - 42*x^3 + 11*x^4)/(1 - x)^7.
Recurrence: a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), n >= 8.
a(n) = (n-1)*(n-2)*(n^4 + 3*n^3 - 20*n^2 - 30*n + 132)/6, n >= 1.
a(n) = A193580(n,3). - R. J. Mathar, Sep 03 2016
E.g.f.: -44 + (1/6)*(264 -264*x +132*x^2 -36*x^3 +38*x^4 +15*x^5 +x^6)*exp(x). - G. C. Greubel, Apr 29 2022

A061997 Number of ways to place 4 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 1, 79, 1987, 16834, 85275, 317471, 962089, 2515262, 5882109, 12605095, 25175191, 47443474, 85152487, 146608359, 243516365, 392004286, 613859609, 938008287, 1402264459, 2055382210, 2959442131, 4192607119

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes, part of V. Kotesovec, Between chessboard and computer, 1996, pp. 204 - 206.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A061995, A061996, A061998, A193580.

R:=PowerSeriesRing(Integers(), 50);
[0,0,0] cat Coefficients(R!( x^3*(1 +70*x +1312*x^2 +1711*x^3 -1209*x^4 -1060*x^5 +1186*x^6 -361*x^7 +30*x^8)/(1-x)^9 )); // G. C. Greubel, Apr 30 2022

CoefficientList[Series[x^3*(1 +70*x +1312*x^2 +1711*x^3 -1209*x^4 -1060*x^5 +1186*x^6 -361*x^7 +30*x^8)/(1-x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,0]+[(n^8 -54*n^6 +72*n^5 +995*n^4 -2472*n^3 -5094*n^2 +21480*n -17112)/24 for n in (3..50)] # G. C. Greubel, Apr 30 2022

G.f.: x^3*(1 + 70*x + 1312*x^2 + 1711*x^3 - 1209*x^4 - 1060*x^5 + 1186*x^6 - 361*x^7 + 30*x^8)/(1 - x)^9.
Recurrence: a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9), n >= 12.
Explicit formula (K.Fabel and K.Soltsien): a(n) = (n^8 - 54*n^6 + 72*n^5 + 995*n^4 - 2472*n^3 - 5094*n^2 + 21480*n - 17112)/24, n >= 3.
a(n) = A193580(n,4). - R. J. Mathar, Sep 03 2016

A172158 Number of ways to place 6 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 978, 62266, 1220298, 12033330, 77784658, 377818258, 1492665418, 5042436754, 15062292834, 40736208186, 101489568538, 235984235970, 517314078210, 1077720399538, 2147500025914, 4114538426818, 7613150953522, 13653752767866, 23808409699242

Offset: 1

Vaclav Kotesovec, Jan 27 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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78, -13,1).

Cf. A061995, A061996, A061997, A061998.

CoefficientList[Series[2*x^4*(489 +24776*x +243562*x^2 +373248*x^3 -287097*x^4 -263140*x^5 +376992*x^6 -162056*x^7 +36103*x^8 -20892*x^9 +14622*x^10 -4432*x^11 +465*x^12)/(1-x)^13, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,0,0,0]+[(n^12 -135*n^10 +180*n^9 +7465*n^8 -18840*n^7 -202665*n^6 + 751860*n^5 +2442334*n^4 -13441200*n^3 -3643800*n^2 +89860320*n -108217440)/720 for n in (5..40)] # G. C. Greubel, Apr 21 2022

a(n) = (n^12 - 135*n^10 + 180*n^9 + 7465*n^8 - 18840*n^7 - 202665*n^6 + 751860*n^5 + 2442334*n^4 - 13441200*n^3 - 3643800*n^2 + 89860320*n - 108217440)/720, n>=5. For any fixed value of k > 1, a(n) = n^(2*k)/k! - 9*n^(2*k-2)/2/(k-2)! + 6*n^(2*k-3)/(k-2)! ... - Vaclav Kotesovec, Jan 27 2010

G.f.: 2*x^5 * (489 + 24776*x + 243562*x^2 + 373248*x^3 - 287097*x^4 - 263140*x^5 + 376992*x^6 - 162056*x^7 + 36103*x^8 - 20892*x^9 + 14622*x^10 - 4432*x^11 + 465*x^12)/(1-x)^13. - Vaclav Kotesovec, Mar 24 2010

More terms from Vincenzo Librandi, May 27 2013

A193580 Triangle read by rows: T(n,k) = number of ways to place k nonattacking kings on an n X n board.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 9, 16, 8, 1, 1, 16, 78, 140, 79, 1, 25, 228, 964, 1987, 1974, 978, 242, 27, 1, 1, 36, 520, 3920, 16834, 42368, 62266, 51504, 21792, 3600, 1, 49, 1020, 11860, 85275, 397014, 1220298, 2484382, 3324193, 2882737, 1601292, 569818, 129657, 18389, 1520, 64, 1

Offset: 0

Andrew Woods, Aug 27 2011

Rows 2n and 2n-1 both contain 1 + n^2 entries. Cf. A008794.
Row n sums to A063443(n+1).
Number of walks of length n-1 on a graph in which each node represents a 11-avoiding n-bit binary sequence B and adjacency of B and B' is determined by B'&(B|(B<<1)|(B>>1))=0 and the total number of nonzero bits in the walk is k.
Row n gives the coefficients of the independence polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017

			The table begins with T(0,0):
  1;
  1,   1;
  1,   4;
  1,   9,  16,   8,   1;
  1,  16,  78, 140,  79;
  ...
T(4,3) = 140 because there are 140 ways to place 3 kings on a 4 X 4 chessboard so that no king threatens any other.

Norman Biggs, Algebraic Graph Theory, Cambridge University Press, New York, NY, second edition, 1993.

Liang Kai, Rows n = 0..26, flattened (Rows n = 0..20 from Andrew Woods, row n = 21 from Alois P. Heinz)
Kai Liang, Independent Set Enumeration in King Graphs by Tensor Network Contractions, arXiv:2505.12776 [math.CO], 2025. See p. 1.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 [math.CO], 2016, Section 4.1.
Johan Nilsson, On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.2.
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, King Graph

Columns 2 to 10: A061995, A061996, A061997, A061998, A172158, A194788, A201369, A201771, A220467.
Diagonal: A201513.
Cf. A179403, etc., for extension to toroidal boards.
Cf. A166540, etc., for extension into three dimensions.
Cf. A098487 for a clipped version.
Row n sums to A063443(n+1).

T(n, 0) = 1;
T(n, 1) = n^2;
T(2n-1, n^2-1) = n^3;
T(2n-1, n^2) = 1.

A201513 Number of ways to place n nonattacking kings on an n X n board.

Original entry on oeis.org

1, 1, 0, 8, 79, 1974, 62266, 2484382, 119138166, 6655170642, 423677826986, 30242576462856, 2390359529372724, 207127434998494421, 19516867860507198208, 1986288643031862123264, 217094567491104327256049, 25357029929230564723578520, 3151672341378566296926684684

Offset: 0

Vaclav Kotesovec, Dec 02 2011

Alois P. Heinz, Table of n, a(n) for n = 0..21 (terms n=1..20 from Andrew Woods)
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 77.

Cf. A061995, A061996, A061997, A061998, A172158, A194788, A201369.
Cf. A063443.
Main diagonal of A098487, A193580.

Asymptotics (Vaclav Kotesovec, Nov 29 2011): n^(2n)/n!*exp(-9/2).

a(0)=1 prepended by Alois P. Heinz, May 11 2017

A172228 Number of ways to place 5 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 1, 304, 10741, 127960, 870589, 4197456, 16005187, 51439096, 145085447, 369074128, 863338777, 1883786680, 3875953561, 7583888944, 14206566327, 25617069208, 44663199283, 75572017136, 124485188701, 200156902936, 314851577749, 485484612496, 735056106571, 1094434774968

Offset: 1

Vaclav Kotesovec, Jan 29 2010

Wazir is a (fairy chess) leaper [0,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
Eric Weisstein's World of Mathematics, Grid Graph
Wikipedia, Fairy chess piece
Wikipedia, Wazir (chess)

Cf. A172225, A172226, A172227, A108792, A061998, A172129, A172136, A172140, A006506.

CoefficientList[Series[x^2 (6 x^11 - 26 x^10 - 93 x^9 + 527 x^8 + 490 x^7 - 6710 x^6 + 13630 x^5 - 3954 x^4 - 26364 x^3 - 7452 x^2 - 293 x - 1) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (n^10-50n^8+40n^7+995n^6-1560n^5-8890n^4+21080n^3+24264n^2-97440n+59520)/120, n>=4.
For any fixed value of k > 1, a(n) = n^(2k)/k! - 5/2/(k-2)!*n^(2k-2) + ...
G.f.: x^3 * (6*x^11 -26*x^10 -93*x^9 +527*x^8 +490*x^7 -6710*x^6 +13630*x^5 -3954*x^4 -26364*x^3 -7452*x^2 -293*x -1) / (x-1)^11. - Vaclav Kotesovec, Apr 29 2011
a(n) = A232833(n,5). - R. J. Mathar, Apr 11 2024

Corrected a(4) and g.f., Vaclav Kotesovec, Apr 29 2011.

More terms from Vincenzo Librandi, May 28 2013

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

A194788 Number of ways to place 7 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 242, 51504, 2484382, 44601420, 450193818, 3112919712, 16471667554, 71393226972, 265069706646, 869583076752, 2577681275622, 7020477731884, 17794428237522, 42397762374912, 95726217156906, 206149749502012, 425731784898894, 846919172059632

Offset: 1

Andrew Woods, Sep 02 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes

Cf. A061995, A061996, A061997, A061998, A172158.

a(n) = (n^14 - 189n^12 + 252n^11 + 15211n^10 - 38640n^9 - 649215n^8 + 2408700n^7 + 14771764n^6 - 75856200n^5 - 144099396n^4 + 1198867488n^3 - 255900576n^2 - 7543005120n + 10617929280)/5040, n>=6. - Andrew Woods, Sep 02 2011

G.f.: 2*x^5*(1930*x^15 - 20052*x^14 + 87663*x^13 - 265681*x^12 + 816798*x^11 - 2117376*x^10 + 2865281*x^9 + 557737*x^8 - 6577818*x^7 + 3848604*x^6 + 8828017*x^5 - 9464319*x^4 - 6316750*x^3 - 868616*x^2 - 23937*x - 121)/ (x-1)^15. - Vaclav Kotesovec, Nov 06 2011

A286440 Number of ways to tile an n X n X n triangular area with five 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-20) of 1 X 1 X 1 tiles.

Original entry on oeis.org

0, 546, 14064, 157248, 1056516, 5086902, 19399860, 62311740, 175452816, 445146906, 1037833944, 2255992584, 4622997276, 9007684494, 16802136156, 30169344996, 52381036968, 88270019922, 144826036032, 231969248016, 363541216308, 558559556262, 842789431428, 1250692671180

Offset: 5

Heinrich Ludwig, May 12 2017

Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061998.

			There are 546 ways of tiling a triangular area of side 6 with 5 tiles of side 2 and an appropriate number (= 16) of tiles of side 1. See illustration in links section.

Heinrich Ludwig, Table of n, a(n) for n = 5..100
Heinrich Ludwig, Illustration of tiling a 6X6X6 area
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A061998, A286436, A286437, A286438, A286439, A286441, A286442.

concat(0, Vec(6*x^6*(91 + 1343*x + 5429*x^2 + 1703*x^3 - 4419*x^4 - 789*x^5 + 2379*x^6 - 627*x^7 - 76*x^8 - 14*x^9 + 20*x^10) / (1 - x)^11 + O(x^40))) \\ Colin Barker, May 12 2017

a(n) = (n^10 -15*n^9 +5*n^8 +930*n^7 -3325*n^6 -19863*n^5 +109915*n^4 +155100*n^3 -1365876*n^2 -191592*n +5981760)/120 for n >= 6.

G.f.: 6*x^6*(91 + 1343*x + 5429*x^2 + 1703*x^3 - 4419*x^4 - 789*x^5 + 2379*x^6 - 627*x^7 - 76*x^8 - 14*x^9 + 20*x^10) / (1 - x)^11. - Colin Barker, May 12 2017

cp's OEIS Frontend

A061995 Number of ways to place 2 nonattacking kings on an n X n board.

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A061996 Number of ways to place 3 nonattacking kings on an n X n board.

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A061997 Number of ways to place 4 nonattacking kings on an n X n board.

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A172158 Number of ways to place 6 nonattacking kings on an n X n board.

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A193580 Triangle read by rows: T(n,k) = number of ways to place k nonattacking kings on an n X n board.

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A201513 Number of ways to place n nonattacking kings on an n X n board.

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A172228 Number of ways to place 5 nonattacking wazirs 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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