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

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

Author

Vaclav Kotesovec, Feb 18 2010

Keywords

• nonn
• nice
• easy

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Christopher R. H. Hanusa, T Zaslavsky, S Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016.
• V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Wikipedia, Nightrider (chess)
• Index entries for linear recurrences with constant coefficients, signature (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).

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