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Number of triangles a queen can make (starting anywhere) on an n X n board.
1

%I #8 Mar 31 2012 13:22:15

%S 0,0,4,28,80,180,332,560,864,1272,1780,2420,3184,4108,5180,6440,7872,

%T 9520,11364,13452,15760,18340,21164,24288,27680,31400,35412,39780,

%U 44464,49532,54940,60760,66944,73568,80580,88060,95952,104340,113164

%N Number of triangles a queen can make (starting anywhere) on an n X n board.

%H Problem of the week, <a href="http://mathforum.org/wagon">Web site</a> - problem 855

%F Harris Kwong (kwong(AT)cs.fredonia.edu): 13 Binomial[ n, 3 ] + 5 Binomial[ n, 2 ] if n is odd or 13 Binomial[ n, 3 ] + 5 Binomial[ n, 2 ] - n/2 if n is even.

%F Contribution from _Sergey Perepechko_, Dec 03 2008: (Start)

%F G.f.: 4*x^2*(2*x^3+5*x^2+5*x+1)/((x - 1)^4*(x + 1)^2).

%F Also a(n)=n*((n-1)*(13*n-8)/6-[n/2]), where [x]=floor(x).

%F Also a(n)+(3*n-1)*binomial(n,3) gives number of cycles of length 3 in a queen's graph associated with this chessboard (see A144298). (End)

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_.

%E More terms from _Erich Friedman_.