A179424
Number of ways to place 4 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 12, 575, 9837, 63553, 265008, 853497, 2312925, 5532967, 12037068, 24293243, 46125317, 83243925, 143918272, 239811333, 387002853, 607226187, 929346700, 1391111127, 2041198973, 2941608713, 4170413232, 5824920625, 8025278157, 10918558863, 14683371948, 19535039827, 25731386325
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).
-
CoefficientList[Series[x^3 (160 x^9 - 963 x^8 + 2054 x^7 - 1308 x^6 - 963 x^5 - 375 x^4 + 5288 x^3 - 5094 x^2 -467 x - 12) / (x - 1)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
A179425
Number of ways to place 5 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 0, 10, 14940, 229908, 1678336, 8155404, 30614620, 96011322, 263506752, 652150382, 1485650012, 3161648520, 6355083264, 12167739256, 22339050588, 39536586430, 67748508480, 112804636266, 183057635420, 290261282204, 450688785408, 686540794500, 1027700020828, 1513897376994, 2197363228480, 3146046781446, 4447496831580
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).
-
CoefficientList[Series[- 2 x^4 (260 x^11 - 1932 x^10 + 6567 x^9 - 16223 x^8 + 38507 x^7 - 77869 x^6 + 102208 x^5 - 61576 x^4 - 15301 x^3 + 33059 x^2 + 7415 x + 5) / (x - 1)^11, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
A179426
Number of ways to place 6 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 0, 0, 10596, 486668, 7063520, 55345356, 299491100, 1263811604, 4455716184, 13701863604, 37823872044, 95648273100, 224887404416, 497181121100, 1042609380588, 2088337713332, 4017815773400, 7459198321428, 13414493857116, 23444476061772, 39928736913120, 66425550447500, 108162598959740, 172697249542932, 270794133842456, 417578468928308, 634036069773900
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
-
CoefficientList[Series[4 x^5 (426 x^13 - 4263 x^12 + 22311 x^11 - 82449 x^10 + 220918 x^9 - 391803 x^8 + 369356 x^7 + 10716 x^6 - 382230 x^5 + 163719 x^4 + 387689 x^3 - 390831 x^2 - 87230 x - 2649) / (x - 1)^13, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
A179427
Number of ways to place 7 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 0, 0, 3420, 576856, 19760512, 270487188, 2209065700, 12914201256, 59659859232, 231216019632, 781647658596, 2367858314700, 6553746728448, 16815788711212, 40446802230372, 92003239814224, 199311860224800, 413589922308360, 825997764087012, 1594007700404532, 2982430581363072, 5425904270482500, 9622254525739492, 16669554533555832, 28264133502586912, 46982453295836640, 76676963241363300
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (15, -105, 455, -1365, 3003, -5005, 6435, -6435, 5005, -3003, 1365, -455, 105, -15, 1).
-
CoefficientList[Series[- 4 x^5 (1379 x^16 - 18219 x^15 + 124755 x^14 - 553765 x^13 + 1657983 x^12 - 3369984 x^11 + 4870575 x^10 - 6400905 x^9 + 10992208 x^8 - 19069951 x^7 + 21246441 x^6 - 8631071 x^5 - 7797385 x^4 + 8273322 x^3 + 2866693 x^2 + 131389 x + 855) / (x - 1)^15, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
A179428
Number of ways to place 8 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 0, 0, 486, 346381, 36285336, 956078397, 12428297150, 104000525596, 643409498286, 3191250652226, 13361641961066, 48905750870775, 160414160371552, 480243686391743, 1330654487994234, 3449609146025210, 8439769551278350, 19624142987739108, 43616849672119790, 93112709811981557, 191696927842663704, 381920049400830625, 738532765420347014, 1389708580432837752, 2550402748009811870, 4573836436177381798, 8029626473495462850
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).
-
CoefficientList[Series[x^5 (17728 x^19 - 301964 x^18 + 2573500 x^17 - 13833040 x^16 + 51521058 x^15 - 143708688 x^14 + 325486412 x^13 - 629393865 x^12 + 996601251 x^11 - 1090603627 x^10 + 426710617 x^9 + 807953488 x^8 - 1328885640 x^7 + 262625618 x^6 + 1106513030 x^5 - 875387697 x^4 - 386005021 x^3 - 30462955 x^2 - 338119 x - 486) / (x - 1)^17, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 01 2013 *)
A194651
Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.
Original entry on oeis.org
0, 0, 0, 88, 785, 3528, 11151, 28560, 63513, 127520, 236863, 413736, 687505, 1096088, 1687455, 2521248, 3670521, 5223600, 7286063, 9982840, 13460433, 17889256, 23466095, 30416688, 38998425, 49503168, 62260191, 77639240, 96053713, 117963960, 143880703
Offset: 1
- V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights
- Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
-
CoefficientList[Series[-x^3*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x - 1)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)
A180067
Number of ways to place 9 nonattacking kings on an n X n toroidal board.
Original entry on oeis.org
0, 0, 0, 0, 0, 28, 81095, 42752576, 2436444603, 53633024900, 666519047964, 5655962632720, 36502953719310, 191587564345044, 854990702601025, 3346890268570368, 11756179090049177, 37692541754516628, 111774885566128630, 309788198526691600
Offset: 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
- Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).
-
CoefficientList[Series[- x^5 (56520 x^22 - 1215064 x^21 + 12642984 x^20 - 82438064 x^19 + 378510176 x^18 - 1315100032 x^17 + 3593010018 x^16 - 7742517098 x^15 + 12798616135 x^14 - 15614945085 x^13 + 14742135008 x^12 - 17197088896 x^11 + 33440162097 x^10 - 55183782403 x^9 + 50601858342 x^8 - 7249042450 x^7 - 32800069391 x^6 + 23010354469 x^5 + 14572795412 x^4 + 1637985772 x^3 + 41216559 x^2 + 80563 x + 28) / (x - 1)^19, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2013 *)
Showing 1-7 of 7 results.