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A124479 From the game of Quod: number of "squares" on an n X n array of points with the four corner points deleted.

%I A124479 #15 Apr 16 2022 15:44:24
%S A124479 0,1,11,37,88,175,311,511,792,1173,1675,2321,3136,4147,5383,6875,8656,
%T A124479 10761,13227,16093,19400,23191,27511,32407,37928,44125,51051,58761,
%U A124479 67312,76763,87175,98611,111136,124817,139723,155925,173496,192511,213047,235183,259000
%N A124479 From the game of Quod: number of "squares" on an n X n array of points with the four corner points deleted.
%C A124479 We count all squares whose vertices are among the points; the sides of the squares need not be horizontal or vertical.
%D A124479 Ian Stewart, How To Cut A Cake: and Other Mathematical Conundrums, Chap. 7.
%H A124479 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A124479 a(n) = (n^4 - n^2 - 48*n + 84)/12.
%F A124479 G.f.: x^3*(1+6*x-8*x^2+3*x^3)/(1-x)^5. [_Colin Barker_, May 21 2012]
%e A124479 So for n=3 we have 5 points:
%e A124479 .....O
%e A124479 ....OOO
%e A124479 .....O
%e A124479 The only square is formed by the 4 outer points, agreeing with a(3)=1.
%e A124479 For n=4 we have 12 points:
%e A124479 .....OO
%e A124479 ....OOOO
%e A124479 ....OOOO
%e A124479 .....OO
%e A124479 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.
%t A124479 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 *)
%K A124479 nonn,easy
%O A124479 2,3
%A A124479 _Joshua Zucker_, Dec 18 2006
%E A124479 Additional comments from _Dean Hickerson_, Dec 18 2006