A190395 Number of ways to place 3 nonattacking grasshoppers on a chessboard of size n x n.

0, 4, 76, 516, 2172, 6860, 17904, 40796, 83976, 159732, 285220, 483604, 785316, 1229436, 1865192, 2753580, 3969104, 5601636, 7758396, 10566052, 14172940, 18751404, 24500256, 31647356, 40452312, 51209300, 64250004, 79946676

Offset: 1

Author

Vaclav Kotesovec, May 10 2011

Keywords

• nonn
• easy

Comments

The Grasshopper moves on the same lines as a queen, but must jump over a hurdle to land on the square immediately beyond.

References

• Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights (in English and Czech)

Crossrefs

Cf. A047659.

Programs

Mathematica

CoefficientList[Series[-4 x (3 x^5 - 7 x^4 + 4 x^3 + 17 x^2 + 12 x + 1) / (x - 1)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 02 2013 *)

Formula

Explicit formula (C. Poisson, 1990): a(n) = 1/6*(n-1)(n^5 +n^4 -2*n^3 -22*n^2 +76*n -72).

G.f.: -4*x^2*(3*x^5 -7*x^4 +4*x^3 +17*x^2 +12*x +1)/(x-1)^7.