A124479 From the game of Quod: number of "squares" on an n X n array of points with the four corner points deleted.
0, 1, 11, 37, 88, 175, 311, 511, 792, 1173, 1675, 2321, 3136, 4147, 5383, 6875, 8656, 10761, 13227, 16093, 19400, 23191, 27511, 32407, 37928, 44125, 51051, 58761, 67312, 76763, 87175, 98611, 111136, 124817, 139723, 155925, 173496, 192511, 213047, 235183, 259000
Offset: 2
Examples
So for n=3 we have 5 points: .....O ....OOO .....O The only square is formed by the 4 outer points, agreeing with a(3)=1. For n=4 we have 12 points: .....OO ....OOOO ....OOOO .....OO There are 5 unit squares, 4 tilted ones with sides sqrt(2) and 2 tilted ones with sides sqrt(5), agreeing with a(4)=11.
References
- Ian Stewart, How To Cut A Cake: and Other Mathematical Conundrums, Chap. 7.
Links
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Programs
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Mathematica
Drop[CoefficientList[Series[x^3(1+6x-8x^2+3x^3)/(1-x)^5,{x,0,50}],x],2] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,11,37,88},50] (* Harvey P. Dale, Apr 16 2022 *)
Formula
a(n) = (n^4 - n^2 - 48*n + 84)/12.
G.f.: x^3*(1+6*x-8*x^2+3*x^3)/(1-x)^5. [Colin Barker, May 21 2012]
Extensions
Additional comments from Dean Hickerson, Dec 18 2006
Comments