A172124 -id:A172124 - cp's OEIS frontend

A172127 Number of ways to place 4 nonattacking bishops on an n X n board.

0, 0, 8, 260, 2728, 16428, 70792, 242856, 706048, 1809464, 4199064, 8992684, 18024072, 34170724, 61784632, 107243472, 179645376, 291667440, 460615272, 709686228, 1069477928, 1579767068, 2291594536, 3269684088, 4595235136

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
Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv:1609.00853 [math.CO], 2016.
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 (6, -12, 2, 27, -36, 0, 36, -27, -2, 12, -6, 1).

Cf. A061994, A172123, A172124.

m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))); // G. C. Greubel, Nov 04 2018

CoefficientList[Series[-4 x^2 (6 x^8 + 57 x^7 + 316 x^6 + 763 x^5 + 1056 x^4 + 791 x^3 + 316 x^2 + 53 x + 2) / ((x-1)^9 (x+1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)
LinearRecurrence[{6,-12,2,27,-36,0,36,-27,-2,12,-6,1},{0,0,8,260,2728,16428,70792,242856,706048,1809464,4199064,8992684},30] (* Harvey P. Dale, Dec 09 2017 *)

x='x+O('x^50); concat([0,0], Vec(4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3))) \\ G. C. Greubel, Nov 04 2018

Explicit formula (Karl Fabel, 1966): a(n) = n(n - 2)(15n^6 - 90n^5 + 260n^4 - 524n^3 + 727n^2 - 646n + 348)/360 if n is even and a(n) = (n - 1)(n - 2)(15n^6 - 75n^5 + 185n^4 - 339n^3 + 388n^2 - 258n + 180)/360 if n is odd.

G.f.: 4*x^3*(6*x^8 +57*x^7 +316*x^6 +763*x^5 +1056*x^4 +791*x^3 +316*x^2 +53*x +2)/((1-x)^9*(x+1)^3). - Vaclav Kotesovec, Mar 25 2010

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

Original entry on oeis.org

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

A172134 Number of ways to place 3 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 4, 36, 276, 1360, 4752, 13340, 32084, 68796, 135040, 247152, 427380, 705144, 1118416, 1715220, 2555252, 3711620, 5272704, 7344136, 10050900, 13539552, 17980560, 23570764, 30535956, 39133580, 49655552, 62431200, 77830324

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 (7,-21,35,-35,21,-7,1).

Cf. A047659, A172124, A172132.

Column k=3 of A244081.

[n le 3 select (n*(n-1))^2 else (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6: n in [1..50]]; // G. C. Greubel, Apr 18 2022

CoefficientList[Series[4x(3x^8 -20x^7 +43x^6 -38x^5 +23x^4 -11x^3 -27x^2 -2x -1)/ (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

def A172134(n):
    if (n<4): return (n*(n-1))^2
    else: return (n-2)*(n+5)*(n^4 -3*n^3 -8*n^2 +66*n -108)/6
[A172134(n) for n in (1..50)] # G. C. Greubel, Apr 18 2022

Explicit formula (Karl Fabel, 1966): a(n) = (n - 2)*(n + 5)*(n^4 - 3*n^3 - 8*n^2 + 66*n - 108)/6, for n >= 4.
G.f.: 4*x^2*(3*x^8-20*x^7+43*x^6-38*x^5+23*x^4-11*x^3-27*x^2-2*x-1)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010
From G. C. Greubel, Apr 18 2022: (Start)
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), for n >= 11.
E.g.f.: (1/6)*(-1080 - 312*x + 12*x^2 +13*x^3 + (1080 - 768*x + 228*x^2 + 38*x^4 + 15*x^5 + x^6)*exp(x)). (End)

A172226 Number of ways to place 3 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 0, 22, 276, 1474, 5248, 14690, 35012, 74326, 144544, 262398, 450580, 739002, 1166176, 1780714, 2642948, 3826670, 5420992, 7532326, 10286484, 13830898, 18336960, 24002482, 31054276, 39750854, 50385248, 63287950

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
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, Wazir (chess)
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A172225, A047659, A061996, A172124, A172134, A172138.

I:=[0, 0, 22, 276, 1474, 5248, 14690, 35012]; [n le 8 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[0] cat [(n-2)*(n^5+2*n^4-11*n^3-10*n^2+42*n-12)/6: n in [2..30]]; // Vincenzo Librandi, Apr 30 2013

A172226:=n->`if`(n=1, 0, (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6); seq(A172226(n), n=1..60); # Wesley Ivan Hurt, Feb 06 2014

CoefficientList[Series[2 x^2 (x^5 - 9 x^4 + 22 x^3 - 2  x^2 - 61 x - 11) / (x-1)^7, {x, 0, 60}], x] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,22,276,1474,5248,14690,35012},30] (* Harvey P. Dale, Apr 08 2022 *)

a(n) = (n-2)*(n^5 + 2*n^4 - 11*n^3 - 10*n^2 + 42*n - 12)/6, n>=2.
G.f.: 2*x^3*(x^5-9*x^4+22*x^3-2*x^2-61*x-11)/(x-1)^7. - Vaclav Kotesovec, Mar 25 2010
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). - Vincenzo Librandi, Apr 30 2013
a(n) = A232833(n,3). - R. J. Mathar, Apr 11 2024

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).

A201244 Number of ways to place 3 non-attacking ferses on an n X n board.

Original entry on oeis.org

0, 0, 38, 340, 1630, 5552, 15210, 35828, 75530, 146240, 264702, 453620, 742918, 1171120, 1786850, 2650452, 3835730, 5431808, 7545110, 10301460, 13848302, 18357040, 24025498, 31080500, 39780570, 50418752, 63325550, 78871988, 97472790, 119589680, 145734802

Offset: 1

Vaclav Kotesovec, Nov 28 2011

Fers is a leaper [1,1].

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, p.415
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A172124, A201243, A201245, A201246, A201247, A201248.

I:=[0, 0, 38, 340, 1630, 5552, 15210, 35828]; [n le 8 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[0] cat [(n-2)*(n^5+2*n^4-11*n^3 +2*n^2+54*n-60)/6: n in [2..35]]; // Vincenzo Librandi, Apr 30 2013

CoefficientList[Series[- 2 x^2 (x^5 + 3 x^4 - 24 x^3 + 24 x^2 + 37 x + 19) / (x-1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Apr 30 2013 *)

a(n) = (n-2)*(n^5 + 2n^4 - 11n^3 + 2n^2 + 54n - 60)/6, n>=2.
G.f.: -2x^3*(x^5 + 3x^4 - 24x^3 + 24x^2 + 37x + 19)/(x-1)^7.
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). - Vincenzo Librandi, Apr 30 2013

A172138 Number of ways to place 3 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 4, 84, 452, 1772, 5596, 14888, 34640, 72712, 140716, 255036, 437968, 718980, 1136092, 1737376, 2582576, 3744848, 5312620, 7391572, 10106736, 13604716, 18056028, 23657560, 30635152, 39246296, 49782956, 62574508, 77990800

Offset: 1

Vaclav Kotesovec, Jan 26 2010

A 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 (7,-21,35,-35,21,-7,1).

Cf. A047659, A172124, A172134, A172137.

[0,4,84,452,1772] cat [(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6: n in [6..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[4x(1+14*x-13*x^2+58*x^3-29*x^4-9*x^5+x^6+ 33*x^7- 45*x^8 +23*x^9-4*x^10)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,4,84,452,1772,5596,14888,34640,72712,140716,255036,437968},30] (* Harvey P. Dale, Mar 11 2023 *)

[0,4,84,452,1772]+[(n^6 -27*n^4 +120*n^3 +74*n^2 -1608*n +2976)/6 for n in (6..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^6 - 27*n^4 + 120*n^3 + 74*n^2 - 1608*n + 2976)/6, n >=6.

G.f.: 4*x^2*(1 + 14*x - 13*x^2 + 58*x^3 - 29*x^4 - 9*x^5 + x^6 + 33*x^7 - 45*x^8 + 23*x^9 - 4*x^10)/(1-x)^7. - Vaclav Kotesovec, Mar 25 2010

A177756 Number of ways to place 3 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 0, 6, 128, 600, 2688, 7350, 19968, 42336, 89600, 163350, 297600, 490776, 809088, 1242150, 1906688, 2774400, 4036608, 5633766, 7862400, 10613400, 14326400, 18818646, 24718848, 31740000

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, 4, -10, -5, 20, 0, -20, 5, 10, -4, -2, 1).

Cf. A172124, A177755.

CoefficientList[Series[- 2 x^2 * (3 x^8 + 58 x^7 + 160 x^6 + 518 x^5 + 442 x^4 + 518 x^3 + 160 x^2 + 58 x + 3)/((x - 1)^7 * (x + 1) ^5), {x, 0,1 50}], x] (* Vincenzo Librandi, May 31 2013 *)
LinearRecurrence[{2,4,-10,-5,20,0,-20,5,10,-4,-2,1},{0,0,6,128,600,2688,7350,19968,42336,89600,163350,297600},30] (* Harvey P. Dale, Aug 31 2024 *)

Explicit formula: 1/12*(n-2)^2*n^2*(2*n^2-4*n+5+3(-1)^n).

G.f.: -2*x^3*(3*x^8+58*x^7+160*x^6+518*x^5+442*x^4+518*x^3+160*x^2+58*x+3)/((x-1)^7*(x+1)^5).

cp's OEIS Frontend

A172127 Number of ways to place 4 nonattacking bishops on an n X n board.

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

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A172134 Number of ways to place 3 nonattacking knights on an n X n board.

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A172226 Number of ways to place 3 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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Mathematica

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A201244 Number of ways to place 3 non-attacking ferses on an n X n board.

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