A177756 -id:A177756 - cp's OEIS frontend

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

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).

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).

A177759 Number of ways to place 6 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 2304, 35280, 811008, 5080320, 38784000, 153679680, 699678720, 2120152320, 7113012480, 18036018000, 49416536064, 110279070720, 261526745088, 530024705280, 1128038400000

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, 10, -22, -44, 110, 110, -330, -165, 660, 132, -924, 0, 924, -132, -660, 165, 330, -110, -110, 44, 22, -10, -2, 1).

Cf. A176886, A177755, A177756, A177757, A177758.

CoefficientList[Series[- 48 x^5 * (15 x^17 + 2386 x^16 + 6778 x^15 + 133898 x^14 + 235216 x^13 + 1520054 x^12 + 1844806 x^11 + 5402462 x^10 + 4378450 x^9 + 6819710 x^8 + 3509350 x^7 + 3079094 x^6 + 926032 x^5 + 445642 x^4 + 65754 x^3 + 14946 x^2 + 639 x + 48) / ((x - 1)^13  (x+1)^11), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)
LinearRecurrence[{2,10,-22,-44,110,110,-330,-165,660,132,-924,0,924,-132,-660,165,330,-110,-110,44,22,-10,-2,1},{0,0,0,0,0,2304,35280,811008,5080320,38784000,153679680,699678720,2120152320,7113012480,18036018000,49416536064,110279070720,261526745088,530024705280,1128038400000,2120098821120,4148067559680,7337013702480,13421018603520},30] (* Harvey P. Dale, Aug 20 2024 *)

Explicit formula: 1/1440*(n-4)^2*(n-2)^2*n^2*(2*n^6 -36*n^5 +269*n^4 -1128*n^3 +3143*n^2-6330*n +7425 +(15*n^4 -240*n^3 +1545*n^2 -4950*n +6975)*(-1)^n).

G.f.: -48*x^6*(15*x^17 +2386*x^16 +6778*x^15 +133898*x^14 +235216*x^13 +1520054*x^12 +1844806*x^11 +5402462*x^10+4378450*x^9 +6819710*x^8 +3509350*x^7 +3079094*x^6+926032*x^5 +445642*x^4 +65754*x^3 +14946*x^2 +639*x+48)/((x-1)^13*(x+1)^11).

A178140 Number of ways to place 7 nonattacking bishops on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 5040, 589824, 6531840, 98304000, 548856000, 3822059520, 14841066240, 67711795200, 208702494000, 726855843840, 1906252508160, 5500708061184, 12796310741760, 32142458880000, 68146033536000

Offset: 1

Vaclav Kotesovec, May 21 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, 12, -26, -65, 156, 208, -572, -429, 1430, 572, -2574, -429, 3432, 0, -3432, 429, 2574, -572, -1430, 429, 572, -208, -156, 65, 26, -12, -2, 1).

Cf. A177755, A177756, A177757, A177758, A177759, A176886.

CoefficientList[Series[- 48 x^6 (105 x^20 + 32558 x^19 + 69284 x^18 + 2532234 x^17 + 4270573 x^16 + 43976860 x^15 + 59687712 x^14 + 262529316 x^13 + 264238506 x^12 + 619225992 x^11 + 438942840 x^10 + 606753672 x^9 + 289183146 x^8 + 243462436 x^7 + 72876832 x^6 + 36501660 x^5 + 6031853 x^4 + 1631114 x^3 + 110244 x^2 + 12078 x + 105) / ((x - 1)^15 (x + 1)^13), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Explicit formula (Vaclav Kotesovec, May 21 2010): (1/10080)*(n-6)^2*(n-4)^2*(n-2)^2*(n^2) * (2*n^6 -36*n^5 +275*n^4 -1224*n^3 +3887*n^2 -9570*n +14625 +(21*n^4 -336*n^3 +2289*n^2 -8190*n +14175)*(-1)^n).

G.f.: -48*x^7 * (105*x^20 +32558*x^19 +69284*x^18 +2532234*x^17 +4270573*x^16 +43976860*x^15 +59687712*x^14 +262529316*x^13 +264238506*x^12 +619225992*x^11 +438942840*x^10 +606753672*x^9 +289183146*x^8 +243462436*x^7 +72876832*x^6 +36501660*x^5 +6031853*x^4 +1631114*x^3 +110244*x^2 +12078*x +105) / ((x-1)^15*(x+1)^13).

A189789 Number of ways to place 8 nonattacking bishops on an n x n toroidal board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 147456, 3265920, 129024000, 1097712000, 12939264000, 66784798080, 436483031040, 1669619952000, 7629571031040, 23828156352000, 85476013572096, 230333593351680, 693478195200000, 1669577821632000

Offset: 1

Vaclav Kotesovec, Apr 27 2011

nonn

Bruno Berselli, Table of n, a(n) for n = 1..5000
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 (2, 14, -30, -90, 210, 350, -910, -910, 2730, 1638, -6006, -2002, 10010, 1430, -12870, 0, 12870, -1430, -10010, 2002, 6006, -1638, -2730, 910, 910, -350, -210, 90, 30, -14, -2, 1).

Cf. A177755, A177756, A177757, A177758, A177759, A178140, A187240

(* Number of ways to place k nonattacking bishops on an n x n toroidal board *)
tbishops[k_,n_]:=If[EvenQ[n],2^k*k!*Sum[Binomial[n/2,i]^2*Binomial[n/2,k-i]^2/Binomial[k,i],{i,0,k}],k!*Binomial[n,k]^2];
Table[tbishops[8,n],{n,1,20}] (* using k=8 for this sequence *)

a(n) = (1/80640) * (n-6)^2 * (n-4)^2 * (n-2)^2 * n^2 * (2*n^8 - 64*n^7 + 884*n^6 - 7048*n^5 + 37382*n^4 - 147904*n^3 + 468540*n^2 - 1108800*n + 1422225 + (28*n^6 - 840*n^5 + 10906*n^4 - 80640*n^3 + 370468*n^2 - 1034880*n + 1400175) * (-1)^n)

G.f.: 1152x^8*(35x^23 + 21178x^22 + 27889x^21 + 2133348x^20 + 3081175x^19 + 51948910x^18 + 72476645x^17 + 469213640x^16 + 538879520x^15 + 1803221880x^14 + 1580004720x^13 + 3146148264x^12 + 2014875632x^11 + 2544618104x^10 + 1144092320x^9 + 933224520x^8 + 278242005x^7 + 143723790x^6 + 25756935x^5 + 7854820x^4 + 693025x^3 + 104538x^2 + 2579x + 128) / ((1-x)^17*(x+1)^15)

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A177757 Number of ways to place 4 nonattacking bishops on an n X n toroidal 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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A177759 Number of ways to place 6 nonattacking bishops on an n X n toroidal board.

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A178140 Number of ways to place 7 nonattacking bishops on an n X n toroidal board.

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A189789 Number of ways to place 8 nonattacking bishops on an n x n toroidal board.

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