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

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

Author

Vaclav Kotesovec, Oct 06 2011

Keywords

• nonn

Comments

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

Links

• V. Kotesovec, Non-attacking chess pieces, 4th edition p.195

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))