A194647   Number of ways to place 5n nonattacking kings on a 10 X 2n cylindrical chessboard
(Vaclav Kotesovec, Sep 7 2011)

G.f.: -2*(7089408*x^21 - 132938496*x^20 + 1125112128*x^19 - 5717239392*x^18 + 19578445344*x^17 - 48082847384*x^16 + 88003026752*x^15 - 123138008952*x^14 + 134072006560*x^13 - 114991853490*x^12 + 78336556962*x^11 - 42596878318*x^10 + 18524447581*x^9 - 6435525481*x^8 + 1778018953*x^7 - 387290192*x^6 + 65568715*x^5 - 8436954*x^4 + 796245*x^3 - 51918*x^2 + 2088*x - 39)/((x-1)*(2*x-1)*(4*x-1)*(6*x-1)*(x^2-4*x+1)*(2*x^2-5*x+1)*(2*x^2-4*x+1)*(4*x^2-6*x+1)*(6*x^2-6*x+1)*(7*x^2-6*x+1)*(2*x^3-8*x^2+6*x-1)*(3*x^3-9*x^2+6*x-1))

Recurrence: a(n) = - 193536*a(n-22) + 4020480*a(n-21) - 37748736*a(n-20) + 213097152*a(n-19) - 811893408*a(n-18) + 2222092032*a(n-17) - 4541105512*a(n-16) + 7111450512*a(n-15) - 8690399936*a(n-14) + 8395031504*a(n-13) - 6469161690*a(n-12) + 4000492482*a(n-11) - 1991743054*a(n-10) + 798883747*a(n-9) - 257594833*a(n-8) + 66416673*a(n-7) - 13565686*a(n-6) + 2162701*a(n-5) - 263028*a(n-4) + 23541*a(n-3) - 1460*a(n-2) + 56*a(n-1)

Explicit formula (in Mathematica format): kings5cyl = (2*6^n + 2*4^n + 8*2^n + 2 + 4*(2+Sqrt[3])^n + 4*(2-Sqrt[3])^n + 2*(2+Sqrt[2])^n + 2*(2-Sqrt[2])^n + 2*((5+Sqrt[17])/2)^n + 2*((5-Sqrt[17])/2)^n + 4*(3+Sqrt[5])^n + 4*(3-Sqrt[5])^n + 4*(3+Sqrt[3])^n + 4*(3-Sqrt[3])^n+4*(3+Sqrt[2])^n + 4*(3-Sqrt[2])^n + 4*(2*Sin[Pi/9])^(2n) + 4*(2*Sin[4Pi/9])^(2n) + 4*(2*Sin[2Pi/9])^(2n) + 4*(2-4/Sqrt[3]*Sin[1/3*ArcSin[3/8*Sqrt[3]]])^n + 4*(2+4/Sqrt[3]*Sin[Pi/3+1/3*ArcSin[3/8*Sqrt[3]]])^n + 4*(2-4/Sqrt[3]*Cos[1/3*ArcCos[-3/8*Sqrt[3]]])^n);