A172529 -id:A172529 - cp's OEIS frontend

A172530 Number of ways to place 3 nonattacking knights on an n X n toroidal board.

0, 0, 6, 208, 600, 3252, 10584, 27584, 61992, 125300, 233772, 409584, 682084, 1089172, 1678800, 2510592, 3657584, 5208084, 7267652, 9961200, 13435212, 17860084, 23432584, 30378432, 38955000, 49454132, 62205084

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172134, A172518.

CoefficientList[Series[- 2 x^2 (50 x^9 - 398 x^8 + 1425 x^7 - 2989 x^6 + 3971 x^5 - 3325 x^4 + 1605 x^3 - 365 x^2 + 83 x + 3) / (x - 1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n^2*(n^4 - 27*n^2 + 218)/6, n>=6.

G.f.: -2*x^3 * (50*x^9 -398*x^8 +1425*x^7 -2989*x^6 +3971*x^5 -3325*x^4 +1605*x^3 -365*x^2 +83*x +3) / (x-1)^7. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 29 2013

A172531 Number of ways to place 4 nonattacking knights on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 228, 600, 12357, 68796, 275888, 872532, 2344025, 5580762, 12107196, 24392446, 46261537, 83426400, 144157632, 240119696, 387393921, 607715342, 929951100

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172530, A172135, A172519.

CoefficientList[Series[x^3 (192 x^13 - 1728 x^12 + 7452 x^11 - 21238 x^10 + 46658 x^9 - 84582 x^8 + 125397 x^7 - 144875 x^6 + 124920 x^5 - 79904 x^4 + 39969 x^3 - 15165 x^2 + 1452 x - 228) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n^2*(n^6 - 54*n^4 + 1115*n^2 - 8934)/24, n>=9.

G.f.: x^4 * (192*x^13 -1728*x^12 +7452*x^11 -21238*x^10 +46658*x^9 -84582*x^8 +125397*x^7 -144875*x^6 +124920*x^5 -79904*x^4 +39969*x^3 -15165*x^2 +1452*x -228) / (x-1)^9. - Vaclav Kotesovec, Mar 25 2010

A172964 Number of ways to place 2 nonattacking knights on an n X n cylindrical board.

Original entry on oeis.org

0, 4, 18, 92, 230, 522, 1022, 1808, 2970, 4610, 6842, 9792, 13598, 18410, 24390, 31712, 40562, 51138, 63650, 78320, 95382, 115082, 137678, 163440, 192650, 225602, 262602, 303968, 350030, 401130, 457622, 519872, 588258, 663170, 745010, 834192, 931142, 1036298, 1150110

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172132.

CoefficientList[Series[2 x (6 x^7 - 30 x^6 + 61 x^5 - 66 x^4 + 45 * x^3 - 21 x^2 + x - 2) / (x - 1)^5, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n*(n^3 - 9*n + 12)/2, n>=5.

G.f.: 2*x^2*(6*x^7-30*x^6+61*x^5-66*x^4+45*x^3-21*x^2+x-2)/(x-1)^5. - Vaclav Kotesovec, Mar 25 2010

More terms from Vincenzo Librandi, May 29 2013

A172532 Number of ways to place 5 nonattacking knights on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 128, 120, 30312, 283906, 1872064, 8643186, 31702920, 98179400, 267487920, 659015500, 1496908840, 3179369070, 6382030592, 12207535134, 22396355496, 39617305308, 67860021680

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172530, A172531, A172136.

CoefficientList[Series[- 2 x^3 (648 x^16 - 10328 x^15 + 71820 x^14 - 295572 x^13 + 818512 x^12 - 1640088 x^11 + 2492742 x^10 - 2967118 x^9 + 2825821 x^8 - 2185007 x^7 + 1376780 x^6 - 677852 x^5 + 219349 x^4 - 32023 x^3 + 18016 x^2 - 644 x + 64) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n^2*(n^8 - 90*n^6 + 3395*n^4 - 64290*n^2 + 522504)/120, n>=10.

G.f.: -2*x^4*(648*x^16-10328*x^15+71820*x^14-295572*x^13+818512*x^12-1640088*x^11+2492742*x^10-2967118*x^9+2825821*x^8-2185007*x^7+1376780*x^6-677852*x^5+219349*x^4-32023*x^3+18016*x^2-644*x+64)/(x-1)^11. - Vaclav Kotesovec, Mar 25 2010

A172533 Number of ways to place 6 nonattacking knights on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 56, 0, 54972, 764596, 8972896, 62560728, 322246800, 1323868260, 4595943336, 14000143196, 38413461800, 96746410800, 226834407552, 500492572112, 1048044384360, 2096986629308, 4031211268200

Offset: 1

Vaclav Kotesovec, Feb 06 2010

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

Cf. A172529, A172530, A172531, A172532.

CoefficientList[Series[4 x^3 (240 x^21 - 3120 x^20 + 20470 x^19 - 105106 x^18 + 512024 x^17 - 2216597 x^16 + 7650408 x^15 - 20251702 x^14 + 41149629 x^13 - 64905350 x^12 + 80399423 x^11 - 78967736 x^10 + 61875645 x^9 - 38631940 x^8 + 19002633 x^7 - 7392461 x^6 + 2560624 x^5 - 840251 x^4 - 8486 x^3 - 14835 x^2 + 182 x - 14) / (x - 1)^13,{x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)

a(n) = n^2*(n^10-135n^8+8005n^6-262665n^4+4816354n^2-39858840)/720, n>=13.

G.f.: 4*x^4*(240*x^21-3120*x^20+20470*x^19-105106*x^18+512024*x^17-2216597*x^16+7650408*x^15-20251702*x^14+41149629*x^13-64905350*x^12+80399423*x^11-78967736*x^10+61875645*x^9-38631940*x^8+19002633*x^7-7392461*x^6+2560624*x^5-840251*x^4-8486*x^3-14835*x^2+182*x-14)/(x-1)^13. - Vaclav Kotesovec, Mar 25 2010

A173436 Number of ways to place 7 nonattacking knights on an n X n toroidal board.

Original entry on oeis.org

0, 0, 0, 16, 0, 80352, 1359288, 31404480, 339256836, 2527519400, 14053530964, 63100177488, 240356217660, 803630856504, 2416671974700, 6655251717376, 17015566051020, 40822003107000, 92679987456312, 200490192134800

Offset: 1

Vaclav Kotesovec, Feb 18 2010

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

Cf. A172529, A172530, A172531, A172532, A172533.

CoefficientList[Series[- 4 x^3 (2535 x^24 - 61497 x^23 + 627330 x^22 - 3849410 x^21 + 16791330 x^20 - 58053150 x^19 + 170691269 x^18 - 438580125 x^17 + 976505385 x^16 - 1844050487 x^15 + 2900976825 x^14 - 3760563305 x^13 + 3991133690 x^12 - 3450574470 x^11 + 2418714751 x^10 - 1370750375 x^9 + 628081926 x^8 - 228075638 x^7 + 56855445 x^6 - 6423333 x^5 + 4868490 x^4 + 36682 x^3 + 20508 x^2 - 60 x + 4) / (x - 1)^15, {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)

Explicit formula: a(n) = n^2*(n^12-189n^10+16135n^8-801255n^6+24595984n^4-445931556n^2+3756080880)/5040, n>=14. For any fixed value of k > 1, a(n) = n^(2k)/k! - 9n^(2k-2)/2/(k-2)! + (243k+143)*n^(2k-4)/24/(k-3)! - ...

G.f.: -4*x^4 * (2535*x^24 -61497*x^23 +627330*x^22 -3849410*x^21 +16791330*x^20 -58053150*x^19 +170691269*x^18 -438580125*x^17 +976505385*x^16 -1844050487*x^15 +2900976825*x^14 -3760563305*x^13 +3991133690*x^12 -3450574470*x^11 +2418714751*x^10 -1370750375*x^9 +628081926*x^8 -228075638*x^7 +56855445*x^6 -6423333*x^5 +4868490*x^4 +36682*x^3 +20508*x^2 -60*x +4) / (x-1)^15. [Vaclav Kotesovec, Mar 25 2010]

cp's OEIS Frontend

A172530 Number of ways to place 3 nonattacking knights on an n X n toroidal board.

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A172531 Number of ways to place 4 nonattacking knights on an n X n toroidal board.

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A172964 Number of ways to place 2 nonattacking knights on an n X n cylindrical board.

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A172532 Number of ways to place 5 nonattacking knights on an n X n toroidal board.

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A172533 Number of ways to place 6 nonattacking knights on an n X n toroidal board.

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A173436 Number of ways to place 7 nonattacking knights on an n X n toroidal board.

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