A172127 -id:A172127 - cp's OEIS frontend

A172129 Number of ways to place 5 nonattacking bishops on an n X n board.

0, 0, 0, 112, 3368, 39680, 282248, 1444928, 5865552, 20014112, 59673360, 159698416, 391202680, 890095584, 1902427800, 3853570560, 7450556064, 13829016768, 24759442464, 42930138864, 72328779720, 118747638592

Offset: 1

Vaclav Kotesovec, Jan 26 2010

For any fixed value of k>1, a(n) = n^(2k) /k! - 2n^(2k - 1) /3/(k - 2)! + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
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 (6,-10,-10,50,-34,-66,110,0,-110,66,34,-50,10,10,-6,1).

Cf. A108792, A172123, A172124, A172127.

Rest[CoefficientList[Series[8*x^4*(14 +337*x +2574*x^2 +9871*x^3 +22040*x^4 +31334*x^5 +28808*x^6 +17522*x^7 +6666*x^8 +1593*x^9 +186*x^10 +15*x^11)/((1-x)^11*(1+x)^5), {x, 0, 50}], x]] (* Vincenzo Librandi, May 02 2013 *)

[(1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(n%2) ) for n in (1..50)] # G. C. Greubel, Apr 17 2022

a(n) = n*(n-2)*(3*n^8 - 34*n^7 + 177*n^6 - 590*n^5 + 1435*n^4 - 2592*n^3 + 3326*n^2 - 2844*n + 1344)/360 if n is even.
a(n) = (n-1)*(n-2)*(n-3)*(3*n^7 - 22*n^6 + 80*n^5 - 204*n^4 + 379*n^3 - 464*n^2 + 378*n - 270)/360 if n is odd.
G.f.: 8*x^4*(14 + 337*x + 2574*x^2 + 9871*x^3 + 22040*x^4 + 31334*x^5 + 28808*x^6 + 17522*x^7 + 6666*x^8 + 1593*x^9 + 186*x^10 + 15*x^11) / ((1-x)^11*(1+x)^5). - Vaclav Kotesovec, Mar 25 2010
a(n) = (1/360)*(n-2)*( n*(1344 -2844*n +3326*n^2 -2592*n^3 +1435*n^4 -590*n^5 +177*n^6 -34*n^7 +3*n^8) -15*(54 -58*n +22*n^2 -3*n^3)*(1-(-1)^n)/2 ). - G. C. Greubel, Apr 17 2022

A172135 Number of ways to place 4 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 1, 18, 412, 4436, 26133, 111066, 376560, 1080942, 2732909, 6253408, 13204356, 26100160, 48819677, 87137934, 149398608, 247349946, 397168485, 620696612, 946921684, 1413726108, 2069939461, 2977725410, 4215337872

Offset: 1

Vaclav Kotesovec, Jan 26 2010

E. Bonsdorff, K. Fabel, O. Riihimaa, Schach und Zahl, 1966, p. 51-63

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 (9,-36,84,-126,126,-84,36,-9,1).

Cf. A061994, A172127, A172132, A172134.

Column k=4 of A244081.

[0,1,18,412,4436] cat [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24: n in [6..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[x*(1 +9*x +286*x^2 +1292*x^3 -345*x^4 +3099*x^5 -5142*x^6 +3606*x^7 -1162*x^8 -390*x^9 +690*x^10 -312*x^11 +48*x^12)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, May 26 2013 *)

[0,1,18,412,4436] + [(n^8 -54*n^6 +144*n^5 +1019*n^4 -5232*n^3 -2022*n^2 +51120*n -77184)/24 for n in (6..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^8 - 54*n^6 + 144*n^5 + 1019*n^4 - 5232*n^3 - 2022*n^2 + 51120*n - 77184)/24, n >= 6. (Karl Fabel, 1966)
G.f.: x^2 * ( 1 + 9*x + 286*x^2 + 1292*x^3 - 345*x^4 +3099*x^5 - 5142*x^6 + 3606*x^7 - 1162*x^8 - 390*x^9 + 690*x^10 - 312*x^11 + 48*x^12) / (1-x)^9. - Vaclav Kotesovec, Mar 25 2010
E.g.f.: x^2/2! + 18*x^3/3! + 412*x^4/4! + 4436*x^5/5! + (1/120)*(385920 + 161040*x + 17940*x^2 - 1200*x^3 - 2660*x^4 - 4484*x^5 + (-385920 + 224880*x - 49860*x^2 + 2940*x^3 + 3250*x^4 + 1920*x^5 + 1060*x^6 + 140*x^7 + 5*x^8)*exp(x)). - G. C. Greubel, Apr 19 2022

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

A176886 Number of ways to place 6 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 16, 1960, 53744, 692320, 5599888, 33001664, 154215760, 603563504, 2052729728, 6229649352, 17202203680, 43870041520, 104531112928, 234870173248, 501360888160

Offset: 1

Vaclav Kotesovec, Apr 28 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Index entries for linear recurrences with constant coefficients, signature (6, -8, -22, 69, -8, -176, 168, 182, -364, 0, 364, -182, -168, 176, 8, -69, 22, 8, -6, 1).

Cf. A172123, A172124, A172127, A172129.

CoefficientList[Series[- 8 x^3 (90 x^15 + 1332 x^14 + 15417 x^13 + 93042 x^12 + 372376 x^11 + 983864 x^10 + 1834807 x^9 + 2423054 x^8 + 2310242 x^7 + 1568260 x^6 + 748519 x^5 + 239742 x^4 + 48236 x^3 + 5264 x^2 + 233 x + 2) / ((x - 1)^13 (x + 1)^7), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

From Vaclav Kotesovec, Apr 27 2010: (Start)
Explicit formula: a(n) = n*(n-2)*(126*n^10 -2268*n^9 +18774*n^8 -97216*n^7 +361165*n^6 -1029454*n^5 +2283178*n^4 -3841960*n^3 +4676932*n^2 -3808152*n +1640160)/90720 if n is even and a(n) = (n-1)*(n-3)*(126*n^10 -2016*n^9 +14868*n^8 -69244*n^7 +234017*n^6 -607984*n^5 +1211879*n^4 -1797328*n^3 +1953593*n^2 -1550820*n +722925)/90720 if n is odd.
G.f.: -8x^4*(90x^15 +1332x^14 +15417x^13 +93042x^12 +372376x^11 +983864x^10 +1834807x^9 +2423054x^8 +2310242x^7 +1568260x^6 +748519x^5 +239742x^4 +48236x^3 +5264x^2 +233x +2)/((x-1)^13*(x+1)^7). (End)

A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

Original entry on oeis.org

1, 2, 9, 70, 729, 9918, 167281, 3423362, 82609921, 2319730026, 74500064809, 2711723081550, 110568316431609, 5016846683306758, 251180326892449969, 13806795579059621930, 827911558468860287041, 53940895144894708523922, 3799498445458163685753481, 288400498147873552894868886

Offset: 0

Vaclav Kotesovec, Dec 06 2011

nonn

Also the number of vertex covers and independent vertex sets of the n X n bishop graph.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320
Vaclav Kotesovec, Non-attacking chess pieces
Eric Weisstein's World of Mathematics, Bishop Graph
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover

Cf. A010790, A172123, A172124, A172127, A172129, A176886, A187239 - A187242, A002465, A216078, A216332.

Row sums of A378590.

knbishops[k_,n_]:=(If[n==1,If[k==1,1,0],(-1)^k/(2n-k)!
*Sum[Binomial[2n-k,n-k+i]*Sum[(-1)^m*Binomial[n-i,m]*m^Floor[n/2]*(m+1)^Floor[(n+1)/2],{m,1,n-i}]
*Sum[(-1)^m*Binomial[n-k+i,m]*m^Floor[(n+1)/2]*(m+1)^Floor[n/2],{m,1,n+i-k}],{i,Max[0,k-n],Min[k,n]}]]);
Table[1+Sum[knbishops[k,n],{k,1,2n-1}],{n,1,25}]

a(n) = A216078(n+1) * A216332(n+1). - Andrew Howroyd, May 08 2017

a(0)=1 prepended by Alois P. Heinz, Dec 01 2024

A187239 Number of ways to place 7 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 440, 38368, 1022320, 14082528, 126490352, 837543200, 4412818240, 19447224864, 74255991784, 251997948736, 774861621936, 2191005028672, 5764306674400, 14243327787456, 33309659739904, 74194554880960, 158241369977880, 324605935279648, 642894402918768

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
E. Weisstein, Bishops Problem, mathWorld.
Index entries for linear recurrences with constant coefficients, signature (6, -6, -34, 84, 42, -322, 162, 603, -708, -540, 1260, 0, -1260, 540, 708, -603, -162, 322, -42, -84, 34, 6, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886.

CoefficientList[Series[- 8 x^4 (630 x^18 + 10620 x^17 + 153525 x^16 + 1211058 x^15 + 6621390 x^14 + 24647178 x^13 + 66958554 x^12 + 133891418 x^11 + 202680754 x^10 + 232634698 x^9 + 204008900 x^8 + 135332502 x^7 + 67245306 x^6 + 24326718 x^5 + 6174582 x^4 + 1024222 x^3 + 99344 x^2 + 4466 x + 55) / ((x - 1)^15 (x + 1)^9), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^14/5040 - n^13/180 + 313n^12/4320 - 383n^11/648 + 14797n^10/4320 - 38233n^9/2520 + 3217n^8/60 - 145469n^7/945 + 1546679n^6/4320 - 4297801n^5/6480 + 257903n^4/270 - 3915679n^3/3780 + 1787007n^2/2240 - 318023n/840 + 9503/128 + (-n^8/192 + n^7/8 - 389n^6/288 + 689n^5/80 - 319n^4/9 + 1153n^3/12 - 95965n^2/576 + 20129n/120 - 9503/128)*(-1)^n.
G.f.: -8x^5*(630x^18 + 10620x^17 + 153525x^16 + 1211058x^15 + 6621390x^14 + 24647178x^13 + 66958554x^12 + 133891418x^11 + 202680754x^10 + 232634698x^9 + 204008900x^8 + 135332502x^7 + 67245306x^6 + 24326718x^5 + 6174582x^4 + 1024222x^3 + 99344x^2 + 4466x + 55)/((x-1)^15*(x+1)^9).
a(7) = A002465(7).

A201245 Number of ways to place 4 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 29, 661, 6285, 35378, 143787, 468529, 1301351, 3202970, 7170593, 14872997, 28969129, 53527866, 94568255, 160741233, 264175507, 421511954, 655152581, 994751765, 1478979173, 2157585442, 3093803379, 4367119121, 6076449375, 8343762538, 11318183177

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A172127, A201243, A201244, A201246, A201247, A201248.

CoefficientList[Series[- x^2 (2 x^8 - 55 x^7 + 230 x^6 - 254 x^5 - 225 x^4 + 173 x^3 + 1380 x^2 + 400 x + 29)/(x-1)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = (n^8 - 30n^6 + 48n^5 + 299n^4 - 912n^3 - 462n^2 + 4368n - 4200)/24, n>=3.

G.f.: -x^3*(2*x^8 - 55*x^7 + 230*x^6 - 254*x^5 - 225*x^4 + 173*x^3 + 1380*x^2 + 400*x + 29)/(x-1)^9.

A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 1, 126, 1168, 7334, 35749, 137970, 438984, 1208246, 2969389, 6662480, 13873100, 27144408, 50389581, 89424014, 152638280, 251834530, 403250693, 628798516, 957543164, 1427453780, 2087456085, 2999819778, 4242915176

Offset: 1

Vaclav Kotesovec, Jan 26 2010

nonn

Zebra is a (fairy chess) leaper [2,3].

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 (9,-36,84,-126,126,-84,36,-9,1).

Cf. A061994, A172127, A172135, A172137, A172138.

CoefficientList[Series[x(1+117*x+70*x^2+1274*x^3+1333*x^4-2109*x^5-462*x^6 +8858*x^7-17006*x^8+15166*x^9-6838*x^10+1478*x^11-650*x^12+760*x^13-376*x^14 +64*x^15)/(1-x)^9, {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,1,126,1168,7334,35749,137970,438984] + [(n^8 -54*n^6 +240*n^5 +827*n^4 -8592*n^3 +10362*n^2 +75600*n -204864)/24 for n in (9..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^8 - 54*n^6 + 240*n^5 + 827*n^4 - 8592*n^3 + 10362*n^2 + 75600*n - 204864)/24, n >= 9.

G.f.: x^2*(1 + 117*x + 70*x^2 + 1274*x^3 + 1333*x^4 - 2109*x^5 - 462*x^6 + 8858*x^7 - 17006*x^8 + 15166*x^9 - 6838*x^10 + 1478*x^11 - 650*x^12 + 760*x^13 - 376*x^14 + 64*x^15)/(1-x)^9. - Vaclav Kotesovec, Mar 25 2010

A177757 Number of ways to place 4 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 64, 600, 6912, 29400, 132864, 381024, 1139200, 2613600, 6177600, 12269400, 24912384, 44717400, 81636352, 135945600, 229423104, 360561024, 572788800, 859685400, 1301766400, 1881864600, 2740725504, 3840540000

Offset: 1

Vaclav Kotesovec, May 13 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
Index entries for linear recurrences with constant coefficients, signature (2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1).

Cf. A172127, A177755, A177756.

CoefficientList[Series[- 8 x^3 (3 x^11 + 122 x^10 + 401 x^9 + 2508 x^8 + 3316 x^7 + 7780 x^6 + 5172 x^5 + 5236 x^4 + 1609 x^3 + 666 x^2 + 59 x + 8)/((x -  1)^9 (x + 1)^7), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)
LinearRecurrence[{2,6,-14,-14,42,14,-70,0,70,-14,-42,14,14,-6,-2,1},{0,0,0,64,600,6912,29400,132864,381024,1139200,2613600,6177600,12269400,24912384,44717400,81636352},50] (* Harvey P. Dale, Nov 05 2016 *)

a(n) = 1/48*(n-2)^2*n^2*(2n^4 -16n^3 +50n^2 -84n +81 +(6n^2 -36n +63)*(-1)^n).

G.f.: -8x^4*(3x^11 +122x^10 +401x^9 +2508x^8 +3316x^7 +7780x^6 +5172x^5 +5236x^4 +1609x^3 +666x^2 +59x+8)/((x-1)^9*(x+1)^7).

A187240 Number of ways to place 8 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 32, 12944, 867328, 22522960, 328097824, 3209594096, 23460698496, 137045115696, 670158151296, 2835083100640, 10634260782464, 36033282628832, 111923478184128, 322412415716896, 869530617762304, 2212626780591008, 5346773160475488, 12336574243905648, 27303885052866048

Offset: 1

Vaclav Kotesovec, Mar 07 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)
E. Weisstein, Bishops Problem, MathWorld
Index entries for linear recurrences with constant coefficients, signature (6, -4, -46, 95, 116, -496, 44, 1331, -990, -2068, 2838, 1683, -4488, 0, 4488, -1683, -2838, 2068, 990, -1331, -44, 496, -116, -95, 46, 4, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886, A187239.

CoefficientList[Series[- 16 x^4 (2520 x^22 + 47160 x^21 + 808884 x^20 + 7825113 x^19 + 54648810 x^18 + 265795497 x^17 + 965510650 x^16 + 2638742416 x^15 + 5598377728 x^14 + 9280070520 x^13 + 12189441400 x^12 + 12689244954 x^11 + 10499675700 x^10 + 6853251794 x^9 + 3501200340 x^8 + 1373620536 x^7 + 404231224 x^6 + 85610168 x^5 + 12313860 x^4 + 1085765 x^3 + 49362 x^2 + 797 x + 2) / ((x - 1)^17 (x + 1)^11), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^16/40320 - n^15/1080 + 7n^14/432 - 1153n^13/6480 + 53951n^12/38880 - 187277n^11/22680 + 106928053n^10/2721600 - 13957093n^9/90720 + 182160427n^8/362880 - 8821499n^7/6480 + 1176831457n^6/388800 - 490477369n^5/90720 + 8235592409n^4/1088640 - 726205757n^3/90720 + 1815275047n^2/302400 - 7953419n/2880 + 8491/16 + (-n^10/960 + 5n^9/144 - 307n^8/576 + 1793n^7/360 - 90571n^6/2880 + 201911n^5/1440 - 513865n^4/1152 + 477841n^3/480 - 4271471n^2/2880 + 1269721n/960 - 8491/16)*(-1)^n.
G.f.: -16x^5*(2520x^22 + 47160x^21 + 808884x^20 + 7825113x^19 + 54648810x^18 + 265795497x^17 + 965510650x^16 + 2638742416x^15 + 5598377728x^14 + 9280070520x^13 + 12189441400x^12 + 12689244954x^11 + 10499675700x^10 + 6853251794x^9 + 3501200340x^8 + 1373620536x^7 + 404231224x^6 + 85610168x^5 + 12313860x^4 + 1085765x^3 + 49362x^2 + 797x + 2)/((x-1)^17*(x+1)^11).
a(8) = A002465(8).

cp's OEIS Frontend

A172129 Number of ways to place 5 nonattacking bishops on an n X n board.

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A172135 Number of ways to place 4 nonattacking knights 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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A176886 Number of ways to place 6 nonattacking bishops on an n X n board.

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A201862 Number of ways to place k nonattacking bishops on an n X n board, sum over all k>=0.

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A187239 Number of ways to place 7 nonattacking bishops on an n X n board.

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A201245 Number of ways to place 4 non-attacking ferses on an n X n board.

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A172139 Number of ways to place 4 nonattacking zebras on an n X n board.

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