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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

We count all squares whose vertices are among the points; the sides of the squares need not be horizontal or vertical.

REFERENCES

Ian Stewart, How To Cut A Cake: and Other Mathematical Conundrums, Chap. 7.

LINKS

Table of n, a(n) for n=2..42.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

(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]

EXAMPLE

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.

CROSSREFS

Sequence in context: A188382 A090950 A217947 * A140373 A316191 A003020

Adjacent sequences:  A124476 A124477 A124478 * A124480 A124481 A124482

KEYWORD

nonn,easy

AUTHOR

Joshua Zucker, Dec 18 2006

EXTENSIONS

Additional comments from Dean Hickerson, Dec 18 2006

STATUS

approved

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Last modified November 18 09:55 EST 2019. Contains 329261 sequences. (Running on oeis4.)