A172129 -id:A172129 - cp's OEIS frontend

A172136 Number of ways to place 5 nonattacking knights on an n X n board.

0, 0, 2, 340, 9386, 97580, 649476, 3184708, 12472084, 41199404, 119171110, 309957412, 739123094, 1639655452, 3422020324, 6778432292, 12833460256, 23356032940, 41051290730, 69954580804

Offset: 1

Vaclav Kotesovec, Jan 26 2010

For any fixed value of k>1, a(n) = n^(2*k) /k! - 9*n^(2*k - 2) /2/(k - 2)! + 12*n^(2*k - 3) /(k - 2)! + ...

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

Cf. A108792, A172129, A172132, A172134, A172135.

Column k=5 of A244081.

CoefficientList[Series[2x^2(74 x^15 -518x^14 +1110x^13 +1046x^12 -11332x^11 + 29950x^10 -42430x^9 +32476x^8 -11684x^7 -1000x^6 +15021x^5 -18443x^4 -6352x^3 - 2878x^2 -159x -1)/(x-1)^11, {x,0,40}], x] (* Vincenzo Librandi, May 02 2013 *)

[0,0,2,340,9386,97580,649476] + [(n^10 -90*n^8 +240*n^7 +3235*n^6 - 16320*n^5 -40530*n^4 +396480*n^3 -231656*n^2 -3359520*n +6509280)/120 for n in (8..50)] # G. C. Greubel, Apr 19 2022

Explicit formula: a(n) = (n^10 - 90*n^8 + 240*n^7 + 3235*n^6 - 16320*n^5 - 40530*n^4 + 396480*n^3 - 231656*n^2 - 3359520*n + 6509280)/120, n >= 8.

G.f.: 2*x^3 * (74*x^15 -518*x^14 +1110*x^13 +1046*x^12 -11332*x^11 +29950*x^10 -42430*x^9 +32476*x^8 -11684*x^7 -1000*x^6 +15021*x^5 -18443*x^4 -6352*x^3 -2878*x^2 -159*x -1) / (x-1)^11. [Vaclav Kotesovec, Mar 25 2010]

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)

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

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).

A201246 Number of ways to place 5 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 12, 780, 16286, 159452, 992412, 4567836, 16959488, 53617596, 149618794, 377841356, 879314442, 1911495356, 3922051616, 7657895196, 14321764860, 25791609308, 44921419134, 75946019596, 125016699158, 200899440924, 315872975684, 486869916572, 736910896536

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

Cf. A172129, A201243, A201244, A201245, A201247, A201248.

CoefficientList[Series[2 x^2 (11 x^11 - 135 x^10 + 549 x^9 - 993 x^8 + 1172 x^7 - 2968 x^6 + 7085 x^5 - 4715x^4 - 10613 x^3 - 4183 x^2- 324 x - 6)/(x-1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = n^10/120 - 5n^8/12 + 2n^7/3 + 191n^6/24 - 24n^5 - 661n^4/12 + 880n^3/3 - 937n^2/15 - 1176n + 1436, n>=4.

G.f.: 2x^3*(11x^11 - 135x^10 + 549x^9 - 993x^8 + 1172x^7 - 2968x^6 + 7085x^5 - 4715x^4 - 10613x^3 - 4183x^2 - 324x - 6)/(x-1)^11.

A172140 Number of ways to place 5 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 0, 126, 2032, 20502, 160696, 929880, 4117520, 15037036, 47368960, 132577826, 336828368, 789558314, 1729320120, 3574328936, 7027309888, 13226773092, 23959787480, 41954706558, 71276149776, 117848892710, 190142197976

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

Cf. A108792, A172129, A172136, A172137, A172138, A172139.

CoefficientList[Series[2x^2(100x^19 -648x^18 +1450x^17 -2126x^16 +10452x^15 - 43872x^14 +92798x^13 -100834x^12 +56460x^11 -61636x^10 +182288x^9 -303224x^8 + 275038x^7 -128982x^6 +21681x^5 +1933x^4 -13072x^3 -2540x^2 -323x-63)/(x-1)^11, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

[0,0,126,2032,20502,160696,929880,4117520,15037036,47368960,132577826] + [(n^10 -90*n^8 +400*n^7 +2915*n^6 -26880*n^5 +2430*n^4 +609920*n^3 - 1517496*n^2 -4188480*n +16581120)/120 for n in (12..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^10 - 90*n^8 + 400*n^7 + 2915*n^6 - 26880*n^5 + 2430*n^4 + 609920*n^3 - 1517496*n^2 - 4188480*n + 16581120)/120, n >= 12.
For any fixed value of k > 1, a(n) = n^(2k) /k! - 9n^(2k - 2) /2/(k - 2)! + 20n^(2k - 3) /(k - 2)! + ...
G.f.: 2*x^3 * (100*x^19 -648*x^18 +1450*x^17 -2126*x^16 +10452*x^15 -43872*x^14 +92798*x^13 -100834*x^12 +56460*x^11 -61636*x^10 +182288*x^9 -303224*x^8 +275038*x^7 -128982*x^6 +21681*x^5 +1933*x^4 -13072*x^3 -2540*x^2 -323*x -63) / (x-1)^11. - Vaclav Kotesovec, Mar 25 2010

A177758 Number of ways to place 5 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 120, 6912, 52920, 466944, 1905120, 8647680, 25613280, 81838080, 198764280, 510478080, 1082161080, 2393997312, 4594961280, 9120190464, 16225246080, 29656350720, 49689816120, 85128088320, 135870624120

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, 8, -18, -27, 72, 48, -168, -42, 252, 0, -252, 42, 168, -48, -72, 27, 18, -8, -2, 1).

Cf. A172129, A177755, A177756, A177757.

CoefficientList[Series[- 24 x^4 (5 x^14 + 406 x^13 + 1333 x^12 + 14880 x^11 + 24307 x^10 + 97498 x^9 + 95187 x^8 + 175328 x^7 + 100307 x^6 + 93018 x^5 + 28147 x^4 + 12832 x^3 + 1589 x^2 + 278 x + 5) / ((x - 1)^11 (x + 1)^9), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Explicit formula: 1/240*(n-4)^2*(n-2)^2*n^2*(2n^4 -16n^3 +54n^2 -108n+153 +(10n^2 -60n +135)*(-1)^n).

G.f.: -24x^5*(5x^14 +406x^13 +1333x^12 +14880x^11 +24307x^10 +97498x^9 +95187x^8 +175328x^7 +100307x^6 +93018x^5 +28147x^4 +12832x^3 +1589x^2 +278x+5)/((x-1)^11*(x+1)^9).

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).

A187241 Number of ways to place 9 nonattacking bishops on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 1600, 389312, 22057472, 565532992, 8611750848, 90564534336, 720227187456, 4603893554496, 24675964279680, 114402835995392, 469601097840640, 1737913582100864, 5882030372643968, 18417596366384512, 53854324059153920, 148209412582029184, 386390343290393024, 959556901097413696

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)
Index entries for linear recurrences with constant coefficients, signature (6, -2, -58, 102, 214, -690, -234, 2418, -962, -5226, 4862, 7150, -11154, -5434, 16302, 0, -16302, 5434, 11154, -7150, -4862, 5226, 962, -2418, 234, 690, -214, -102, 58, 2, -6, 1).

Cf. A172123, A172124, A172127, A172129, A176886, A187239, A187240.

CoefficientList[Series[- 64 x^5 (5670 x^25 + 116100 x^24 + 2282283 x^23 + 25883910 x^22 + 220244661 x^21 + 1330673229 x^20 + 6121839129 x^19 + 21511823232 x^18 + 59645434477 x^17 + 131494649245 x^16 + 234424379246 x^15 + 339339084372 x^14 + 401937236082 x^13 + 389328811002 x^12 + 308645316626 x^11 + 199052247464 x^10 + 103780570480 x^9 + 43151321222 x^8 + 14078209111 x^7 + 3508317590 x^6 + 644755881 x^5 + 82579449 x^4 + 6782181 x^3 + 308200 x^2 + 5933 x + 25) / ((x - 1)^19 (x + 1)^13), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

a(n) = n^18/362880 - n^17/7560 + 181n^16/60480 - 14509n^15/340200 + 2101n^14/4860 - 101071n^13/30240 + 112406401n^12/5443200 - 143351879n^11/1360800 + 2465350549n^10/5443200 - 14081834n^9/8505 + 55888723201n^8/10886400 - 6055816813n^7/453600 + 155816526107n^6/5443200 - 13489156949n^5/272160 + 183801705823n^4/2721600 - 15816472541n^3/226800 + 30820237351n^2/604800 - 919392091n/40320 + 1101239/256 + (-n^12/5760 + 11n^11/1440 - 113n^10/720 + 51793n^9/25920 - 202873n^8/11520 + 3428791n^7/30240 - 1050169n^6/1920 + 8590259n^5/4320 - 1034689n^4/192 + 68481311n^3/6480 - 81534479n^2/5760 + 465686363n/40320 - 1101239/256)*(-1)^n.
G.f.: -64x^6*(5670x^25 + 116100x^24 + 2282283x^23 + 25883910x^22 + 220244661x^21 + 1330673229x^20 + 6121839129x^19 + 21511823232x^18 + 59645434477x^17 + 131494649245x^16 + 234424379246x^15 + 339339084372x^14 + 401937236082x^13 + 389328811002x^12 + 308645316626x^11 + 199052247464x^10 + 103780570480x^9 + 43151321222x^8 + 14078209111x^7 + 3508317590x^6 + 644755881x^5 + 82579449x^4 + 6782181x^3 + 308200x^2 + 5933x + 25)/((x-1)^19*(x+1)^13).
a(9) = A002465(9).

cp's OEIS Frontend

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A172228 Number of ways to place 5 nonattacking wazirs 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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A201246 Number of ways to place 5 non-attacking ferses on an n X n board.

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A172140 Number of ways to place 5 nonattacking zebras on an n X n board.

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A177758 Number of ways to place 5 nonattacking bishops on an n X n toroidal board.

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