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.

Previous Showing 11-16 of 16 results.

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

Views

Author

Vaclav Kotesovec, Jan 07 2011

Keywords

Links

Crossrefs

Cf. A061998, A179403, A179404, A179424.

Programs

  • Mathematica
    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 *)

Formula

Explicit formula: a(n) = 1/120*n^2*(n^8-90n^6+3155n^4-51450n^2+332544), n>=6.
G.f.: -2x^5*(260x^11 - 1932x^10 + 6567x^9 - 16223x^8 + 38507x^7 - 77869x^6 + 102208x^5 - 61576x^4 - 15301x^3 + 33059x^2 + 7415x + 5)/(x-1)^11.

A172204 Number of ways to place 5 nonattacking kings on a 5 X n board.

Original entry on oeis.org

0, 0, 15, 194, 1974, 9856, 34475, 95466, 224589, 468854, 893646, 1585850, 2656976, 4246284, 6523909, 9693986, 13997775, 19716786, 27175904, 36746514, 48849626, 63959000, 82604271, 105374074, 132919169
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 29 2010

Keywords

Links

Crossrefs

Cf. A061998, A061991, A172202, A172203.

Programs

  • Mathematica
    CoefficientList[Series[x^2 * (259 * x^6 - 204 * x^5 + 1294 * x^4 + 622 * x^3 + 1035 * x^2 + 104 * x + 15)/(x - 1)^6, {x, 0, 40}], x] (* Vincenzo Librandi, May 27 2013 *)

Formula

a(n) = (625n^5-9750n^4+66415n^3-247626n^2+504664n-446544)/24, n>=4.
G.f.: x^3*(259*x^6-204*x^5+1294*x^4+622*x^3+1035*x^2+104*x+15)/(x-1)^6. - Vaclav Kotesovec, Mar 24 2010

A201369 Number of ways to place 8 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 27, 21792, 3324193, 119138166, 1979541332, 20142680752, 145977165234, 824771174978, 3850985758339, 15461577137802, 54912339921707, 176153338628674, 518569625849418, 1418340918023792, 3639736652346172, 8833161922947702, 20405252721413369
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 30 2011

Keywords

Links

Crossrefs

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788.

Formula

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Nov 30 2011): (n^16 - 252*n^14 + 336*n^13 + 27762*n^12 - 70896*n^11 - 1699656*n^10 + 6330240*n^9 + 60677169*n^8 - 304864560*n^7 - 1181816748*n^6 + 8314366704*n^5 + 8495481308*n^4 - 121101870624*n^3 + 74007948336*n^2 + 730891869120*n - 1180990460160)/40320, n>=7.
G.f.: -x^5*(14882*x^18 - 180784*x^17 + 1061244*x^16 - 4500406*x^15 + 15038864*x^14 - 34328850*x^13 + 40903004*x^12 - 8667835*x^11 + 23857551*x^10 - 260744627*x^9 + 545801251*x^8 - 276255996*x^7 - 467674682*x^6 + 484515328*x^5 + 391528458*x^4 + 65572237*x^3 + 2957401*x^2 + 21333*x + 27)/(x-1)^17.

A201771 Number of ways to place 9 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 1, 3600, 2882737, 229095676, 6655170642, 103395053720, 1051588999820, 7878155295948, 46838274976147, 232322652402464, 995789500001315, 3784235129731708, 12999197522073908, 40969826999523768, 119876498636101786, 328726265508168780, 851369417500529061
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 04 2011

Keywords

Links

Crossrefs

Cf. A061995, A061996, A061997, A061998, A172158, A193580, A194788, A201369.

Formula

Explicit formula (Vaclav Kotesovec, after values computed by Andrew Woods, Dec 04 2011): n^18/362880 - n^16/1120 + n^15/840 + 1559*n^14/12096 - 119*n^13/360 - 7681*n^12/720 + 479*n^11/12 + 9383677*n^10/17280 - 195031*n^9/72 - 24176483*n^8/1440 + 4447749*n^7/40 + 5032857271*n^6/18144 - 495178813*n^5/180 - 2551293629*n^4/2520 + 1588223225*n^3/42 - 11469403819*n^2/315 - 664490248*n/3 + 405670140, n>=8.
G.f.: x^5*(54764*x^21 - 805588*x^20 + 6061268*x^19 - 31485512*x^18 + 117971558*x^17 - 312791986*x^16 + 620038858*x^15 - 1193322246*x^14 + 2685590901*x^13 - 4918483903*x^12 + 3824558880*x^11 + 5110355848*x^10 - 13987162841*x^9 + 5213745395*x^8 + 15789867458*x^7 - 14255103822*x^6 - 13342741937*x^5 - 2791816301*x^4 - 174938304*x^3 - 2814508*x^2 - 3581*x - 1)/(x-1)^19.

A220467 Number of ways to place 10 nonattacking kings on an n X n board.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1601292, 314949564, 17143061738, 423677826986, 6210264633994, 62831788827614, 481992723228798, 2982908737810114, 15548436178142582, 70420082692285198, 283631426534134042, 1034163399690010346, 3461457325296584554, 10754832937513676198
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 15 2012

Keywords

Links

Crossrefs

Cf. A061995 (2 kings), A061996 (3 kings), A061997 (4 kings).
Cf. A061998 (5 kings), A172158 (6 kings), A194788 (7 kings).
Cf. A201369 (8 kings), A201771 (9 kings).
Column k=10 of A193580.

Programs

  • Mathematica
    Rest[CoefficientList[Series[-2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21, {x, 0, 20}], x]]

Formula

a(n) = n^20/3628800 - n^18/8960 + n^17/6720 + 353*n^16/17280 - 53*n^15/1008 - 29467*n^14/13440 + 11867*n^13/1440 + 25901053*n^12/172800 - 107495*n^11/144 - 8467959*n^10/1280 + 122792641*n^9/2880 + 32499630031*n^8/181440 - 112903333*n^7/72 - 16042907329*n^6/6720 + 36445613711*n^5/1008 - 1784819159*n^4/300 - 9997453897*n^3/21 + 85979117831*n^2/140 + 13635070421*n/5 - 5609601346, for n>=9.
G.f.: -2*x^7*(97581*x^22 - 1758956*x^21 + 16320562*x^20 - 100734462*x^19 + 443795293*x^18 - 1471049082*x^17 + 3971393292*x^16 - 9304893422*x^15 + 17917931016*x^14 - 22612415810*x^13 + 6949925614*x^12 + 21430418050*x^11 + 9738010368*x^10 - 153051533038*x^9 + 256884162558*x^8 - 71451647970*x^7 - 265785285277*x^6 + 220345759446*x^5 + 251887022384*x^4 + 63841610284*x^3 + 5432696107*x^2 + 140661216*x + 800646)/(x-1)^21.

A194653 Number of ways to place 5 nonattacking kings on an n X n cylindrical chessboard.

Original entry on oeis.org

0, 0, 0, 0, 655, 26952, 309869, 1998752, 9124848, 33065040, 101473009, 274593648, 673080928, 1522931256, 3224953725, 6458355776, 12330557912, 22588294464, 39908439249, 68290845520, 113579839128, 184145882536, 291764365485, 452734505952, 689287992800
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 31 2011

Keywords

Links

Crossrefs

Cf. A061998, A179425, A194650, A194651, A194652.

Formula

a(n) = 1/120*n*(n^9 - 90*n^7 + 60*n^6 + 3155*n^5 - 3900*n^4 - 50910*n^3 + 86580*n^2 + 318864*n - 656160), n>=6.
G.f.: -x^5*(185*x^11 - 1635*x^10 + 6336*x^9 - 15496*x^8 + 32185*x^7 - 62315*x^6 + 86237*x^5 - 49559*x^4 - 35522*x^3 + 49422*x^2 + 19747*x + 655)/(x-1)^11.
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