A061997 -id:A061997 - 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

A061998 Number of ways to place 5 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 1974, 42368, 397014, 2326320, 10087628, 35464464, 106783320, 285336128, 693331146, 1558986816, 3286192514, 6558317232, 12488282352, 22829958032, 40269324564, 68817690624, 114333609854, 185205015936

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A061995, A061996, A061997, A193580.

CoefficientList[Series[2*x^5*(987 +10327*x +19768*x^2 -18152*x^3 -2711*x^4 +5149*x^5 +1774*x^6 -2882*x^7 +958*x^8 -98*x^9)/(1-x)^11, {x,0,45}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,0,0]+[(n-4)*(n^9 +4*n^8 -74*n^7 -176*n^6 +2411*n^5 +1844*n^4 -38194*n^3 +18944*n^2 +236520*n -316320)/120 for n in (4..50)] # G. C. Greubel, May 01 2022

G.f.: 2*x^5*(987 + 10327*x + 19768*x^2 - 18152*x^3 - 2711*x^4 + 5149*x^5 + 1774*x^6 - 2882*x^7 + 958*x^8 - 98*x^9)/(1 - x)^11.
Recurrence: a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11), n >= 15.
Explicit formula (V.Kotesovec, 1992): a(n) = (n - 4)*(n^9 + 4*n^8 - 74*n^7 - 176*n^6 + 2411*n^5 + 1844*n^4 - 38194*n^3 + 18944*n^2 + 236520*n - 316320)/120, n >= 4.
a(n) = A193580(n,5). - 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.

A172227 Number of ways to place 4 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 6, 405, 5024, 31320, 133544, 446421, 1258590, 3126724, 7042930, 14669709, 28658436, 53069000, 93909924, 159819965, 262913874, 419816676, 652912510, 991835749, 1475233800, 2152832664, 3087838016, 4359706245, 6067321574, 8332617060, 11304678954

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Brazeal Slides on a Chessboard, Math Horizons, Vol. 27, pp. 24-27, April 2020.
Vaclav 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, A061994, A061997, A172127, A172135, A172139, A006506.

CoefficientList[Series[- x^2 (4 x^8 - 26 x^7 + 3 x^6 + 303 x^5 - 736 x^4 + 180 x^3 + 1595 x^2 + 351 x + 6) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

a(n) = (n^8-30n^6+24n^5+323n^4-504n^3-1110n^2+2760n-1224)/24, n>=3.
G.f.: -x^3*(4*x^8-26*x^7+3*x^6+303*x^5-736*x^4+180*x^3+1595*x^2+351*x+6)/(x-1)^9. - Vaclav Kotesovec, Apr 29 2011
a(n) = A232833(n,4). - R. J. Mathar, Apr 11 2024

Corrected a(3) and g.f., Vaclav Kotesovec, Apr 29 2011

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

A179424 Number of ways to place 4 nonattacking kings on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 12, 575, 9837, 63553, 265008, 853497, 2312925, 5532967, 12037068, 24293243, 46125317, 83243925, 143918272, 239811333, 387002853, 607226187, 929346700, 1391111127, 2041198973, 2941608713, 4170413232, 5824920625, 8025278157, 10918558863, 14683371948, 19535039827, 25731386325

Offset: 1

Vaclav Kotesovec, Jan 07 2011

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
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A061997, A179403, A179404.

CoefficientList[Series[x^3 (160 x^9 - 963 x^8 + 2054 x^7 - 1308 x^6 - 963 x^5 - 375 x^4 + 5288 x^3 - 5094 x^2 -467 x - 12) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)

Explicit formula: a(n) = 1/24*n^2*(n^6-54*n^4+1019*n^2-6798), n>=5.

G.f.: x^4*(160*x^9 - 963*x^8 + 2054*x^7 - 1308*x^6 - 963*x^5 - 375*x^4 + 5288*x^3 - 5094*x^2 - 467*x - 12)/(x-1)^9.

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

Original entry on oeis.org

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

A286439 Number of ways to tile an n X n X n triangular area with four 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-16) of 1 X 1 X 1 tiles.

Original entry on oeis.org

0, 1, 25, 747, 7459, 42983, 176373, 575775, 1595487, 3908979, 8701313, 17936083, 34713675, 63739327, 111921149, 189119943, 309074343, 490526475, 758575017, 1146284219, 1696579123, 2464458903, 3519561925, 4949117807, 6861323439, 9389181603, 12694842513, 16974490275

Offset: 3

Heinrich Ludwig, May 11 2017

Rotations and reflections of tilings are counted. If they are to be ignored, see A286446. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061997.

			There are 25 ways of tiling a triangular area of side 5 with 4 tiles of side 2 and an appropriate number (= 9) of tiles of side 1. See example in links section.

Heinrich Ludwig, Table of n, a(n) for n = 3..100
Heinrich Ludwig, Illustration of tiling a 5X5X5 area
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A286436, A286446, A286437, A286438, A286440, A286441, A286442, A061997.

concat(0, Vec(x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9 + O(x^60))) \\ Colin Barker, May 12 2017

a(n) = (n^8 -12*n^7 +6*n^6 +432*n^5 -1279*n^4 -4692*n^3 +20592*n^2 +13320*n -91800)/24, for n>=5.

G.f.: x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9. - 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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A061998 Number of ways to place 5 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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A172227 Number of ways to place 4 nonattacking wazirs 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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A179424 Number of ways to place 4 nonattacking kings on an n X n toroidal board.

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