A172123 -id:A172123 - cp's OEIS frontend

A172124 Number of ways to place 3 nonattacking bishops on an n X n board.

0, 0, 26, 232, 1124, 3896, 10894, 26192, 56296, 110960, 204130, 355000, 589196, 940072, 1450134, 2172576, 3172944, 4530912, 6342186, 8720520, 11799860, 15736600, 20711966, 26934512, 34642744, 44107856, 55636594, 69574232

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, and S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853, a12016
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,-14,14,0,-14,14,-6,1).

Cf. A047659, A172123.

[(n*(n-2)*(2*n^4 -4*n^3 +7*n^2 -6*n +4) +3*(n mod 2))/12: n in [1..40]]; // G. C. Greubel, Apr 16 2022

CoefficientList[Series[2x^2(3x^4 +18x^3 +48x^2 +38x +13)/((1-x)^7 (x+1)), {x, 0, 30}], x] (* Vincenzo Librandi, May 26 2013 *)

[(n*(n-2)*(2*n^4 -4*n^3 +7*n^2 -6*n +4) +3*(n%2))/12 for n in (1..40)] # G. C. Greubel, Apr 16 2022

Explicit formulas (Karl Fabel, 1966): (Start)
a(n) = n*(n-2)*(2*n^4 - 4*n^3 + 7*n^2 - 6*n + 4)/12 if n is even.
a(n) = (n-1)*(2*n^5 - 6*n^4 + 9*n^3 - 11*n^2 + 5*n - 3)/12 if n is odd. (End)
G.f.: 2*x^3*(13+38*x+48*x^2+18*x^3+3*x^4)/((1-x)^7*(1+x)). - .Vaclav Kotesovec, Mar 25 2010
a(n) = (2*(n-2)*n*(2*n^4-4*n^3+7*n^2-6*n+4)-3*(-1)^n+3)/24. - Bruno Berselli, May 26 2013
E.g.f.: (1/24)*( (3 - 6*x + 6*x^2 + 100*x^3 + 130*x^4 + 44*x^5 + 4*x^6)*exp(x) - 3*exp(-x) ). - G. C. Greubel, Apr 16 2022

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

Original entry on oeis.org

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

A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

Original entry on oeis.org

0, 2, 24, 96, 260, 570, 1092, 1904, 3096, 4770, 7040, 10032, 13884, 18746, 24780, 32160, 41072, 51714, 64296, 79040, 96180, 115962, 138644, 164496, 193800, 226850, 263952, 305424, 351596, 402810, 459420, 521792, 590304

Offset: 1

Vaclav Kotesovec, Jan 29 2010

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

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p. 829.

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 (5,-10,10,-5,1).

Cf. A036464, A061995, A172132, A172123, A172137.

I:=[0, 2, 24, 96, 260]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[n*(n-1)*(n^2+n-4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[n (n - 1) (n^2 + n - 4) / 2, {n, 40}] (* Vincenzo Librandi, Apr 30 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{0,2,24,96,260},40] (* Harvey P. Dale, Jun 04 2023 *)

Explicit formula (Christian Poisson, 1990): a(n) = n*(n-1)*(n^2+n-4)/2.
G.f.: 2*x^2*(2*x^2-7*x-1)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Apr 30 2013
a(n) = 2*A239352(n). - R. J. Mathar, Jan 09 2018
a(n) = A232833(n,2). - R. J. Mathar, Apr 11 2024

A172132 Number of ways to place 2 nonattacking knights on an n X n board.

Original entry on oeis.org

0, 6, 28, 96, 252, 550, 1056, 1848, 3016, 4662, 6900, 9856, 13668, 18486, 24472, 31800, 40656, 51238, 63756, 78432, 95500, 115206, 137808, 163576, 192792, 225750, 262756, 304128, 350196, 401302, 457800, 520056, 588448, 663366

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
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 (5,-10,10,-5,1).

Cf. A036464, A172123.

Column k=2 of A244081.

I:=[0, 6, 28, 96, 252]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

[(n-1)*(n+4)*(n^2-3*n+4)/2: n in [1..40]]; // Vincenzo Librandi, Apr 30 2013

Table[(n-1)(n+4)(n^2 -3n +4)/2, {n, 40}] (* Vincenzo Librandi, Apr 30 2013 *)

[(n-1)*(n+4)*(n^2-3*n+4)/2 for n in (1..40)] # G. C. Greubel, Apr 18 2022

a(n) = (n - 1)*(n + 4)*(n^2 - 3*n + 4)/2.
G.f.: 2*(12*x^4-39*x^3+37*x^2-20*x+4)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Vincenzo Librandi, Apr 30 2013
E.g.f.: (1/2)*(16 + (-16 + 16*x - 2*x^2 + 6*x^3 + x^4)*exp(x)). - G. C. Greubel, Apr 18 2022

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)

A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

Original entry on oeis.org

0, 6, 28, 96, 240, 518, 980, 1712, 2784, 4310, 6380, 9136, 12688, 17206, 22820, 29728, 38080, 48102, 59964, 73920, 90160, 108966, 130548, 155216, 183200, 214838, 250380, 290192, 334544, 383830, 438340, 498496, 564608, 637126

Offset: 1

Vaclav Kotesovec, Jan 26 2010

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

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, a12016
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 (3,-1,-5,5,1,-3,1).

Cf. A036464, A172123, A172132, A172137.

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)): n in [1..40]]; // G. C. Greubel, Apr 21 2022

CoefficientList[Series[2*x*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2), {x,0,40}], x] (* Vincenzo Librandi, May 27 2013 *)

[(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)) for n in (1..40)] # G. C. Greubel, Apr 21 2022

Explicit formula (Christian Poisson, 1990): a(n) = n*(3*n^3 - 5*n^2 + 9*n - 4)/6 if n is even and a(n) = n*(n - 1)*(3*n^2 - 2*n + 7)/6 if n is odd.
G.f.: 2*x^2*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2). - Vaclav Kotesovec, Mar 25 2010
From G. C. Greubel, Apr 21 2022: (Start)
a(n) = (1/12)*n*(3*(-1)^n - (11 - 18*n + 10*n^2 - 6*n^3)).
E.g.f.: (x/12)*(-3*exp(-x) + (3 + 30*x + 26*x^2 + 6*x^3)exp(x)). (End)

A177755 Number of ways to place 2 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 4, 18, 80, 200, 468, 882, 1600, 2592, 4100, 6050, 8784, 12168, 16660, 22050, 28928, 36992, 46980, 58482, 72400, 88200, 106964, 128018, 152640, 180000

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, 2, -6, 0, 6, -2, -2, 1).

Cf. A172123.

Table[(n^2 (2n^2-4n+3+(-1)^n))/4,{n,30}] (* or *) LinearRecurrence[ {2,2,-6,0,6,-2,-2,1},{0,4,18,80,200,468,882,1600},30] (* Harvey P. Dale, Mar 06 2013 *)
CoefficientList[Series[- 2 x (x^5 + 8 x^4 + 14 x^3 + 18 x^2 + 5 x + 2) / ((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Explicit formula: 1/4*n^2*(2*n^2-4*n+3+(-1)^n).
G.f.: -2*x^2*(x^5+8*x^4+14*x^3+18*x^2+5*x+2)/((x-1)^5*(x+1)^3).
a(1)=0, a(2)=4, a(3)=18, a(4)=80, a(5)=200, a(6)=468, a(7)=882, a(8)=1600, a(n)=2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8). - Harvey P. Dale, Mar 06 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

A172137 Number of ways to place 2 nonattacking zebras on an n X n board.

Original entry on oeis.org

0, 6, 36, 112, 276, 582, 1096, 1896, 3072, 4726, 6972, 9936, 13756, 18582, 24576, 31912, 40776, 51366, 63892, 78576, 95652, 115366, 137976, 163752, 192976, 225942, 262956, 304336, 350412, 401526, 458032, 520296, 588696, 663622, 745476, 834672, 931636, 1036806

Offset: 1

Vaclav Kotesovec, Jan 26 2010

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

Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p. 829.

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 (5,-10,10,-5,1).

Cf. A036464, A172123, A172132.

[n eq 1 select 0 else (n^4 -9*n^2 +40*n -48)/2: n in [1..50]]; // G. C. Greubel, Apr 19 2022

CoefficientList[Series[2x(3+3*x-4*x^2+8*x^3-4*x^4)/(1-x)^55, {x, 0, 40}], x] (* Vincenzo Librandi, May 26 2013 *)

[(n^4 -9*n^2 +40*n -48 +16*bool(n==1))/2 for n in (1..50)] # G. C. Greubel, Apr 19 2022

a(n) = (n^4 - 9*n^2 + 40*n - 48)/2, n >= 2. (Christian Poisson, 1990)
G.f.: 2*x^2*(3+3*x-4*x^2+8*x^3-4*x^4)/(1-x)^5. - Vaclav Kotesovec, Mar 25 2010
E.g.f.: (1/2)*(16*(3+x) + (-48 + 32*x - 2*x^2 + 6*x^3 + x^4)*exp(x)). - G. C. Greubel, Apr 19 2022

cp's OEIS Frontend

A172124 Number of ways to place 3 nonattacking bishops on an n X n board.

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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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A172225 Number of ways to place 2 nonattacking wazirs on an n X n board.

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A172132 Number of ways to place 2 nonattacking knights 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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A172141 Number of ways to place 2 nonattacking nightriders on an n X n board.

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