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 A196812 Number of ways to place 2 nonattacking nightriders on an n X n toroidal board 1

%I #8 Sep 12 2015 11:00:27

%S 0,2,18,72,200,378,588,1312,2106,3650,4840,7848,10140,14210,20250,

%T 25728,32368,42282,51984,67400,80262,97042,116380,141984,167500,

%U 195026,228906,266952,306124,358650,403620,463360,524898,592450,671300,754920,837828,936434

%N Number of ways to place 2 nonattacking nightriders on an n X n toroidal board

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

%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 4th edition p.195

%F 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)

%F 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)

%F 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))

%t 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}]

%Y Cf. A172141, A196810.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Oct 06 2011

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Last modified May 29 08:58 EDT 2023. Contains 363029 sequences. (Running on oeis4.)