

A126275


Moment of inertia of all magic squares of order n.


2



5, 60, 340, 1300, 3885, 9800, 21840, 44280, 83325, 147620, 248820, 402220, 627445, 949200, 1398080, 2011440, 2834325, 3920460, 5333300, 7147140, 9448285, 12336280, 15925200, 20345000, 25742925, 32284980, 40157460, 49568540, 60749925, 73958560, 89478400
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OFFSET

2,1


COMMENTS

"This is the only property of magic squares, aside from the line sums, which is solely dependent on the order of the square, n," Loly and Adam Rogers note in a paper published in 2004 in the Canadian Undergraduate Physics Journal.
Loly investigated the "physical" properties of magic squares, treating the numbers of each such square as physical quantities. If the integers are consecutive numbers from 1 to n^2, the square is said to be of nth order. The magic sum itself is given by n(n^2 + 1)/2.
Suppose you interpret the numbers as masses. You can then determine a magic square's moment of inertia about a given axis of rotation. For any specific case, you obtain the moment of inertia, In, of a magic square of order n about an axis at right angles to its center by summing mr^2 for each cell, where m is the number centered in a cell and r is the distance of the center of that cell from the center of the square measured in units of the nearest neighbor distance.
You find that the moment of inertia, I_z, about the square's center (an axis at right angles to the square) is twice the moment of inertia about an axis of rotation along the center row or column.


LINKS

Table of n, a(n) for n=2..32.
Peter Loly, The invariance of the moment of inertia of magic squares, Mathematical Gazette 88(March 2004):151153
Ivars Peterson, Magic Square Physics. Science News online, Jul 01, 2006; Vol. 170, No. 1
Index entries for linear recurrences with constant coefficients, signature (7,21,35,35,21,7,1).


FORMULA

a(n) = (n^2 * (n^4  1))/12.
G.f.: 5*x^2*(x+1)*(x^2+4*x+1) / (x1)^7. [Colin Barker, Dec 10 2012]


CROSSREFS

Sequence in context: A091457 A289724 A100906 * A059602 A290747 A212700
Adjacent sequences: A126272 A126273 A126274 * A126276 A126277 A126278


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Dec 23 2006


EXTENSIONS

More terms from Colin Barker, Dec 10 2012


STATUS

approved



