A061997 -id:A061997 - cp's OEIS frontend

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

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

A172203 Number of ways to place 4 nonattacking kings on a 4 X n board.

Original entry on oeis.org

0, 0, 9, 79, 454, 1566, 4103, 9009, 17484, 30984, 51221, 80163, 120034, 173314, 242739, 331301, 442248, 579084, 745569, 945719, 1183806, 1464358, 1792159, 2172249, 2609924

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

Cf. A061997, A061990, A172202.

CoefficientList[Series[- x^2 (68 x^4 - 4 x^3 + 149 x^2 + 34 x + 9) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 27 2013 *)

a(n) = (64n^4 - 720n^3 + 3347n^2 - 7569n + 6894)/6, n>=3.

G.f.: -x^3*(68*x^4-4*x^3+149*x^2+34*x+9)/(x-1)^5. - Vaclav Kotesovec, Mar 24 2010

A201369 Number of ways to place 8 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 27, 21792, 3324193, 119138166, 1979541332, 20142680752, 145977165234, 824771174978, 3850985758339, 15461577137802, 54912339921707, 176153338628674, 518569625849418, 1418340918023792, 3639736652346172, 8833161922947702, 20405252721413369

Offset: 1

Vaclav Kotesovec, Nov 30 2011

nonn

V. Kotesovec, Non-attacking chess pieces

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788.

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Nov 30 2011): (n^16 - 252*n^14 + 336*n^13 + 27762*n^12 - 70896*n^11 - 1699656*n^10 + 6330240*n^9 + 60677169*n^8 - 304864560*n^7 - 1181816748*n^6 + 8314366704*n^5 + 8495481308*n^4 - 121101870624*n^3 + 74007948336*n^2 + 730891869120*n - 1180990460160)/40320, n>=7.

G.f.: -x^5*(14882*x^18 - 180784*x^17 + 1061244*x^16 - 4500406*x^15 + 15038864*x^14 - 34328850*x^13 + 40903004*x^12 - 8667835*x^11 + 23857551*x^10 - 260744627*x^9 + 545801251*x^8 - 276255996*x^7 - 467674682*x^6 + 484515328*x^5 + 391528458*x^4 + 65572237*x^3 + 2957401*x^2 + 21333*x + 27)/(x-1)^17.

A194652 Number of ways to place 4 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 32, 1205, 13260, 74494, 291708, 908973, 2416410, 5711530, 12327414, 24743693, 46797968, 84216990, 145288600, 241697109, 389546478, 610597338, 933745570, 1396771845, 2048393204, 2950649438, 4181658708, 5838778525, 8042209890, 10939084074, 14708073198

Offset: 1

Vaclav Kotesovec, Aug 31 2011

nonn

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A061997, A179424, A194650, A194651.

CoefficientList[Series[x^3*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

a(n) = 1/24*n*(n^7 - 54*n^5 + 36*n^4 + 1019*n^3 - 1236*n^2 - 6690*n + 10884), n>=5.

G.f.: x^4*(54*x^9 - 384*x^8 + 1052*x^7 - 1263*x^6 + 657*x^5 - 1434*x^4 + 4154*x^3 - 3567*x^2 - 917*x - 32)/(x-1)^9.

A201771 Number of ways to place 9 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 1, 3600, 2882737, 229095676, 6655170642, 103395053720, 1051588999820, 7878155295948, 46838274976147, 232322652402464, 995789500001315, 3784235129731708, 12999197522073908, 40969826999523768, 119876498636101786, 328726265508168780, 851369417500529061

Offset: 1

Vaclav Kotesovec, Dec 04 2011

nonn

V. Kotesovec, Non-attacking chess pieces

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788, A201369.

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Dec 04 2011): n^18/362880 - n^16/1120 + n^15/840 + 1559*n^14/12096 - 119*n^13/360 - 7681*n^12/720 + 479*n^11/12 + 9383677*n^10/17280 - 195031*n^9/72 - 24176483*n^8/1440 + 4447749*n^7/40 + 5032857271*n^6/18144 - 495178813*n^5/180 - 2551293629*n^4/2520 + 1588223225*n^3/42 - 11469403819*n^2/315 - 664490248*n/3 + 405670140, n>=8.

G.f.: x^5*(54764*x^21 - 805588*x^20 + 6061268*x^19 - 31485512*x^18 + 117971558*x^17 - 312791986*x^16 + 620038858*x^15 - 1193322246*x^14 + 2685590901*x^13 - 4918483903*x^12 + 3824558880*x^11 + 5110355848*x^10 - 13987162841*x^9 + 5213745395*x^8 + 15789867458*x^7 - 14255103822*x^6 - 13342741937*x^5 - 2791816301*x^4 - 174938304*x^3 - 2814508*x^2 - 3581*x - 1)/(x-1)^19.

A220467 Number of ways to place 10 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1601292, 314949564, 17143061738, 423677826986, 6210264633994, 62831788827614, 481992723228798, 2982908737810114, 15548436178142582, 70420082692285198, 283631426534134042, 1034163399690010346, 3461457325296584554, 10754832937513676198

Offset: 1

Vaclav Kotesovec, Dec 15 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces

Cf. A061995 (2 kings), A061996 (3 kings), A061997 (4 kings).
Cf. A061998 (5 kings), A172158 (6 kings), A194788 (7 kings).
Cf. A201369 (8 kings), A201771 (9 kings).
Column k=10 of A193580.

Rest[CoefficientList[Series[-2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21, {x, 0, 20}], x]]

a(n) = n^20/3628800 - n^18/8960 + n^17/6720 + 353*n^16/17280 - 53*n^15/1008 - 29467*n^14/13440 + 11867*n^13/1440 + 25901053*n^12/172800 - 107495*n^11/144 - 8467959*n^10/1280 + 122792641*n^9/2880 + 32499630031*n^8/181440 - 112903333*n^7/72 - 16042907329*n^6/6720 + 36445613711*n^5/1008 - 1784819159*n^4/300 - 9997453897*n^3/21 + 85979117831*n^2/140 + 13635070421*n/5 - 5609601346, for n>=9.

G.f.: -2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21.

cp's OEIS Frontend

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

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A172203 Number of ways to place 4 nonattacking kings on a 4 X n board.

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A201369 Number of ways to place 8 nonattacking kings on an n X n board.

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A194652 Number of ways to place 4 nonattacking kings on an n X n cylindrical chessboard.

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A201771 Number of ways to place 9 nonattacking kings on an n X n board.

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A220467 Number of ways to place 10 nonattacking kings on an n X n board.

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