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A124479
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From the game of Quod: number of "squares" on an n X n array of points with the four corner points deleted.
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0
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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
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OFFSET
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2,3
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COMMENTS
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We count all squares whose vertices are among the points; the sides of the squares need not be horizontal or vertical.
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REFERENCES
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Ian Stewart, How To Cut A Cake: and Other Mathematical Conundrums, Chap. 7.
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LINKS
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FORMULA
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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]
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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