A030117 Number of triangles a queen can make (starting anywhere) on an n X n board.

0, 0, 4, 28, 80, 180, 332, 560, 864, 1272, 1780, 2420, 3184, 4108, 5180, 6440, 7872, 9520, 11364, 13452, 15760, 18340, 21164, 24288, 27680, 31400, 35412, 39780, 44464, 49532, 54940, 60760, 66944, 73568, 80580, 88060, 95952, 104340, 113164

Offset

0,3

Links

Table of n, a(n) for n=0..38.

Problem of the week, Web site - problem 855

Formula

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.

Contribution from Sergey Perepechko, Dec 03 2008: (Start)

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

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

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)

Crossrefs

Sequence in context: A085024 A085001 A153784 * A361173 A005634 A183485

Adjacent sequences: A030114 A030115 A030116 * A030118 A030119 A030120

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Erich Friedman.

Status

approved