cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Vaclav Kotesovec, Jan 07 2011

Keywords

Links

Crossrefs

Cf. A179403, A179404, A179424, A179425, A179426.

Programs

  • Mathematica
    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 *)

Formula

Explicit formula: a(n) = 1/5040*n^2*(n^12 -189*n^10 +15295*n^8 -681135*n^6 +17692024*n^4 -255655596*n^2 +1617230880), n>=8.
G.f.: -4*x^6*(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.

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

Views

Author

Vaclav Kotesovec, Jan 07 2011

Keywords

Links

Crossrefs

Cf. A179403, A179404, A179424, A179425, A179426, A179427.

Programs

  • Mathematica
    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 *)

Formula

a(n) = 1/40320*n^2 * (n^14 -252*n^12 +27874*n^10 -1759800*n^8 +68745649*n^6 -1669136028*n^4 +23447322156*n^2 -147931524720), n>=9.
G.f.: x^6*(17728x^19 - 301964x^18 + 2573500x^17 - 13833040x^16 + 51521058x^15 - 143708688x^14 + 325486412x^13 - 629393865x^12 + 996601251x^11 - 1090603627x^10 + 426710617x^9 + 807953488x^8 - 1328885640x^7 + 262625618x^6 + 1106513030x^5 - 875387697x^4 - 386005021x^3 - 30462955x^2 - 338119x - 486)/(x-1)^17.

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

Views

Author

Vaclav Kotesovec, Jan 15 2011

Keywords

Links

Crossrefs

Cf. A179403, A179404, A179424, A179425, A179426, A179427, A179428.

Programs

  • Mathematica
    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 *)

Formula

a(n) = 1/362880*n^2 * (n^16 -324*n^14 +46914*n^12 -3975048*n^10 +216203169*n^8 -7756575876*n^6 +179987135516*n^4 -2481599151792*n^2 +15651056776320), n>=10.
G.f.: -x^6*(56520x^22 - 1215064x^21 + 12642984x^20 - 82438064x^19 + 378510176x^18 - 1315100032x^17 + 3593010018x^16 - 7742517098x^15 + 12798616135x^14 - 15614945085x^13 + 14742135008x^12 - 17197088896x^11 + 33440162097x^10 - 55183782403x^9 + 50601858342x^8 - 7249042450x^7 - 32800069391x^6 + 23010354469x^5 + 14572795412x^4 + 1637985772x^3 + 41216559x^2 + 80563x + 28)/(x-1)^19.
General asymptotic formula for number of ways to place k nonattacking kings on an n X n toroidal board: n^2k/k! - 9/2*n^(2k-2)/(k-2)! + (243k+47)*n^(2k-4)/(24*(k-3)!) - (243k^2+141k+80)*n^(2k-6)/(16*(k-4)!) + (98415k^3+114210k^2+140645k+101762)*n^(2k-8)/(5760*(k-5)!)-...

A194654 Number of ways to place 6 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 0, 125, 28930, 809368, 9414152, 66305781, 338374560, 1378426060, 4751038284, 14388638901, 39296604844, 98605016040, 230507248912, 507379370525, 1060395103800, 2118303772332, 4066797540820, 7537144196589, 13535598398916, 23628635128024, 40203393674520
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Links

Crossrefs

Cf. A172158, A179426, A194650, A194651, A194652, A194653.

Formula

a(n) = 1/720*n*(n^11 - 135*n^9 + 90*n^8 + 7525*n^7 - 9420*n^6 - 216045*n^5 + 378090*n^4 + 3192694*n^3 - 6899520*n^2 - 19450080*n + 48327120), n>=7.
G.f.: x^5*(622*x^14 - 6966*x^13 + 37088*x^12 - 130876*x^11 + 344918*x^10 - 655255*x^9 + 737997*x^8 - 153262*x^7 - 639936*x^6 + 251910*x^5 + 1132096*x^4 - 1113158*x^3 - 443028*x^2 - 27305*x - 125)/(x-1)^13.
Showing 1-4 of 4 results.

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