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 1-7 of 7 results.

A173429 Number of ways to place 3 nonattacking nightriders on an n X n board.

Original entry on oeis.org

0, 4, 36, 276, 1152, 3920, 10568, 25348, 53848, 106292, 194732, 339416, 562652, 899796, 1388008, 2083908, 3044992, 4356344, 6102144, 8404204, 11380564, 15199100, 20019856, 26067112, 33551812, 42766092, 53981600, 67570804
Offset: 1

Views

Author

Vaclav Kotesovec, Feb 18 2010

Keywords

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

Crossrefs

Cf. A172141, A047659, A172134.

Programs

  • Mathematica
    CoefficientList[Series[-(36 x^29 + 124 x^28 + 496 x^27 + 1128 x^26 + 2632 x^25 + 4280 x^24 + 7160 x^23 + 9296 x^22 + 12936 x^21 + 14828 x^20 + 18828 x^19 + 20164 x^18 + 23820 x^17 + 23684 x^16 + 25460 x^15 + 22972 x^14 + 22412 x^13 + 18532 x^12 + 16820 x^11 + 12996 x^10 + 10912 x^9 + 7552 x^8 + 5428 x^7 + 3012 x^6 + 1652 x^5 + 604 x^4 + 204 x^3 + 28 x^2 + 4 x) / ((x + 1)^4 (x - 1)^7 (x^2 + 1) (x^2 + x + 1) (x^8 + x^6 + x^4 + x^2 + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 30 2013 *)
LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,0,-3,0,2,0,4,-4,0,-2,0,3,0,0,-1,0,-2,2,0,1,0,-2,1},{0,4,36,276,1152,3920,10568,25348,53848,106292,194732,339416,562652,899796,1388008,2083908,3044992,4356344,6102144,8404204,11380564,15199100,20019856,26067112,33551812,42766092,53981600,67570804,83876732,103365728,126463668},30] (* Harvey P. Dale, Dec 27 2015 *)

Formula

a(n) = 1/6*n^6-5/6*n^5+4031/1440*n^4-621/100*n^3+3313/288*n^2-2623/150*n+82321/43200 + (1/4*n^3-25/32*n^2+77/50*n-43/64)*(-1)^n - (1+(-1)^n)/8*cos(Pi*n/2) + 8/27*(-1)^n*cos(Pi*n/3) + (-4*(-1)^n+(sqrt(5)+3+(1-sqrt(5)/5)*(-1)^n)*n)*4/25*cos(Pi*n/5) + (sqrt(58*sqrt(5)+130)-sqrt(50-22*sqrt(5))*(-1)^n/5)*16/25*sin(Pi*n/5) + (-4+(sqrt(5)/5+1+(3-sqrt(5))*(-1)^n)*n)*4/25*cos(2*Pi*n/5) + (sqrt(22*sqrt(5)+50)/5-sqrt(130-58*sqrt(5))*(-1)^n)*16/25*sin(2*Pi*n/5).
Recurrence: a(n) = 2*a(n-1)-a(n-3)-2*a(n-5)+2*a(n-6)+a(n-8)-3*a(n-11)+2*a(n-13)+4*a(n-15)-4*a(n-16)-2*a(n-18)+3*a(n-20)-a(n-23)-2*a(n-25)+2*a(n-26)+a(n-28)-2*a(n-30)+a(n-31), n>=32.
G.f.: -(36*x^30+124*x^29+496*x^28+1128*x^27+2632*x^26+4280*x^25+7160*x^24+9296*x^23+12936*x^22+14828*x^21+18828*x^20+20164*x^19+23820*x^18+23684*x^17+25460*x^16+22972*x^15+22412*x^14+18532*x^13+16820*x^12+12996*x^11+10912*x^10+7552*x^9+5428*x^8+3012*x^7+1652*x^6+604*x^5+204*x^4+28*x^3+4*x^2)/((x+1)^4*(x-1)^7*(x^2+1)*(x^2+x+1)*(x^8+x^6+x^4+x^2+1)^2). - Vaclav Kotesovec, Mar 22 2010

A172218 Number of ways to place 3 nonattacking nightriders on a 3 X n board.

Original entry on oeis.org

1, 12, 36, 100, 213, 408, 712, 1148, 1745, 2528, 3524, 4760, 6263, 8060, 10178, 12644, 15485, 18728, 22400, 26528, 31139, 36260, 41918, 48140, 54953, 62384, 70460, 79208, 88655, 98828, 109754, 121460, 133973, 147320, 161528, 176624, 192635
Offset: 1

Views

Author

Vaclav Kotesovec, Jan 29 2010

Keywords

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

Crossrefs

Cf. A172141, A061989, A172212.

Programs

  • Mathematica
    CoefficientList[Series[(2 x^10 - 4 x^9 + 6 x^8 - 4 x^7 - 6 x^6 + 24 x^5 - 18 x^4 + 24 x^3 - 6 x^2 + 8 x + 1) / (x - 1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 28 2013 *)

Formula

a(n) = (9n^3 - 57n^2 + 210n - 344)/2, n>=8.
G.f.: x*(2*x^10-4*x^9+6*x^8-4*x^7-6*x^6+24*x^5-18*x^4+24*x^3-6*x^2+8*x+1)/(x-1)^4. - Vaclav Kotesovec, Mar 25 2010

A190393 Number of ways to place n nonattacking nightriders on an n X n toroidal board.

Original entry on oeis.org

1, 2, 6, 24, 120, 144, 28, 1408, 2025, 86400, 1782, 1092096, 4186, 31360, 241920000, 23953408, 140692, 114108912, 1092690
Offset: 1

Views

Author

Vaclav Kotesovec, May 10 2011

Keywords

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

Crossrefs

Cf. A172141, A173429, A051906.

Extensions

Terms a(16)-a(17) from Vaclav Kotesovec, May 14 2011
Terms a(18)-a(19) from Vaclav Kotesovec, May 28 2011

A190394 Maximum number of nonattacking nightriders on an n X n board.

Original entry on oeis.org

1, 4, 5, 8, 10, 16, 17, 20, 21, 24, 26, 32, 33, 36, 39, 42, 45, 48, 51, 54, 58, 64, 65, 66, 68, 72, 75, 80, 81, 84, 87, 90, 93
Offset: 1

Views

Author

Vaclav Kotesovec, May 10 2011

Keywords

Comments

A nightrider is a fairy chess piece that can move any distance in a direction specified by a knight move.
Maximum number of nonattacking nightriders on an n X n toroidal board is n.

Examples

			From _Rob Pratt_, Jul 24 2015: (Start)
a(20) = 54:
  XX--XXXX---X------XX
  XX---------X--XX--XX
  --------------------
  ---X----------------
  X-----------------X-
  X-----------------X-
  X-------------------
  X---------X---------
  ------------------XX
  ------------X-------
  -------X------------
  XX------------------
  ---------X---------X
  -------------------X
  -X-----------------X
  -X-----------------X
  ----------------X---
  --------------------
  XX--XX--X---------XX
  XX------X---XXXX--XX
(End)

Links

Crossrefs

Cf. A085801, A190393, A172141, A173429.

Formula

2n <= a(n) <= 3n-2, for n > 3.
a(n) >= 24*floor((n+4)/10)-8, for n >= 6. - Vaclav Kotesovec, Apr 01 2012

Extensions

Terms a(11)-a(16) from Vaclav Kotesovec, May 13 2011
Terms a(17)-a(19) from Vaclav Kotesovec, Apr 01 2012
a(20) from Rob Pratt, Jul 24 2015
a(21)-a(32) from Paul Tabatabai, Nov 06 2018
a(33) from Andy Huchala, Mar 30 2024

A196810 Number of ways to place 2 nonattacking nightriders on an n X n cylindrical board.

Original entry on oeis.org

0, 4, 18, 80, 200, 420, 756, 1472, 2358, 3860, 5500, 8304, 11232, 15484, 21090, 27392, 34816, 44604, 55404, 69840, 84294, 102124, 122452, 147264, 173800, 203476, 237762, 276752, 318304, 368340, 418500, 478208, 541398, 611524, 689780, 774576, 863136, 963148
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 06 2011

Keywords

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

Crossrefs

Cf. A172141, A196812.

Programs

  • Mathematica
    Table[(143*n^2)/30-(79*n^3)/15+n^4/2+16/5*n^2*Floor[n/5]+n^2*Floor[n/4]+4/3*n^2*Floor[n/3]+(n+2*n^2)*Floor[n/2]+8/5*n^2*Floor[(1+n)/5]+n^2*Floor[(1+n)/4]+2/3*n^2*Floor[(1+n)/3]+8/5*n^2*Floor[(2+n)/5]+8/5*n^2*Floor[(3+n)/5],{n,1,100}]

Formula

G.f.: -(2*x^2*(2 + 17*x + 96*x^2 + 384*x^3 + 1203*x^4 + 3100*x^5 + 6917*x^6 + 13670*x^7 + 24466*x^8 + 39974*x^9 + 60206*x^10 + 83709*x^11 + 107667*x^12 + 128088*x^13 + 141070*x^14 + 143882*x^15 + 136037*x^16 + 119239*x^17 + 96892*x^18 + 72808*x^19 + 50428*x^20 + 31926*x^21 + 18321*x^22 + 9388*x^23 + 4223*x^24 + 1622*x^25 + 514*x^26 + 127*x^27 + 22*x^28 + 2*x^29))/((-1+x)^5*(1+x)^3*(1+x^2)^3*(1+x+x^2)^3*(1+x+x^2+x^3+x^4)^3).
Recurrence: a(n) = a(n-32) + 4*a(n-31) + 10*a(n-30) + 17*a(n-29) + 20*a(n-28) + 11*a(n-27) - 15*a(n-26) - 54*a(n-25) - 90*a(n-24) - 99*a(n-23) - 63*a(n-22) + 18*a(n-21) + 116*a(n-20) + 188*a(n-19) + 194*a(n-18) + 123*a(n-17) - 123*a(n-15) - 194*a(n-14) - 188*a(n-13) - 116*a(n-12) - 18*a(n-11) + 63*a(n-10) + 99*a(n-9) + 90*a(n-8) + 54*a(n-7) + 15*a(n-6) - 11*a(n-5) - 20*a(n-4) - 17*a(n-3) - 10*a(n-2) - 4*a(n-1).
Explicit formula: a(n) = -n/4+(572*n^2)/225-(3*n^3)/2+n^4/2+(-1)^n*(n/4+n^2/2)+1/2*n^2*cos((n*Pi)/2)+16/25*n^2*cos((4*n*Pi)/5)+4/9*n^2*cos((4*n*Pi)/3)+16/25*n^2*cos((8*n*Pi)/5).
Chaiken et al. give a 4th degree quasi-polynomial formula. - N. J. A. Sloane, Feb 16 2013
Note that cited formula is for normal chessboard (not cylindrical), see sequence A172141. - Vaclav Kotesovec, Dec 09 2013

A196812 Number of ways to place 2 nonattacking nightriders on an n X n toroidal board.

Original entry on oeis.org

0, 2, 18, 72, 200, 378, 588, 1312, 2106, 3650, 4840, 7848, 10140, 14210, 20250, 25728, 32368, 42282, 51984, 67400, 80262, 97042, 116380, 141984, 167500, 195026, 228906, 266952, 306124, 358650, 403620, 463360, 524898, 592450, 671300, 754920, 837828, 936434
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 06 2011

Keywords

Comments

A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

Links

Crossrefs

Cf. A172141, A196810.

Programs

  • Mathematica
    Table[n^2/2*(21-22*n+n^2+16*Floor[n/5]+4*Floor[n/4]+8*Floor[n/3]+8*Floor[n/2]+8*Floor[(1+n)/5]+4*Floor[(1+n)/4]+4*Floor[(1+n)/3]+8*Floor[(2+n)/5]+8*Floor[(3+n)/5]),{n,1,100}]

Formula

G.f. (Vaclav Kotesovec, Apr 18 2010): -(2*x^2*(2*x^29 + 25*x^28 + 151*x^27 + 620*x^26 + 1965*x^25 + 5094*x^24 + 11169*x^23 + 21370*x^22 + 36349*x^21 + 56009*x^20 + 78898*x^19 + 102778*x^18 + 124128*x^17 + 139254*x^16 + 144792*x^15 + 139276*x^14 + 123618*x^13 + 101232*x^12 + 76538*x^11 + 53680*x^10 + 35008*x^9 + 21359*x^8 + 12037*x^7 + 6226*x^6 + 2853*x^5 + 1122*x^4 + 351*x^3 + 82*x^2 + 13*x + 1))/((x-1)^5*(x+1)^3*(x^2+1)^3*(x^2+x+1)^3*(x^4+x^3+x^2+x+1)^3)
Recurrence: a(n) = a(n-32) + 4*a(n-31) + 10*a(n-30) + 17*a(n-29) + 20*a(n-28) + 11*a(n-27) - 15*a(n-26) - 54*a(n-25) - 90*a(n-24) - 99*a(n-23) - 63*a(n-22) + 18*a(n-21) + 116*a(n-20) + 188*a(n-19) + 194*a(n-18) + 123*a(n-17) - 123*a(n-15) - 194*a(n-14) - 188*a(n-13) - 116*a(n-12) - 18*a(n-11) + 63*a(n-10) + 99*a(n-9) + 90*a(n-8) + 54*a(n-7) + 15*a(n-6) - 11*a(n-5) - 20*a(n-4) - 17*a(n-3) - 10*a(n-2) - 4*a(n-1)
Explicit formula: a(n) = n^2/2*(119/15+2*(-1)^n-4*n+n^2+2*cos((n*Pi)/2) +16/5*cos((4*n*Pi)/5)+8/3*cos((4*n*Pi)/3)+16/5*cos((8*n*Pi)/5))

A196814 Number of ways to place n nonattacking nightriders on an n X n cylindrical board.

Original entry on oeis.org

1, 4, 6, 84, 120, 784, 280, 40816, 13806, 1361706, 110990, 142633176, 4263454, 197730660, 9246172320
Offset: 1

Views

Author

Vaclav Kotesovec, Oct 06 2011

Keywords

Links

Crossrefs

Cf. A190393, A196810, A196811, A172141, A173429
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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