

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
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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*x8*x^2+3*x^3)/(1x)^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



