A126275 Moment of inertia of all magic squares of order n.
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
Offset: 2
Links
- Michael De Vlieger, Table of n, a(n) for n = 2..10000
- Peter Loly, The Invariance of the Moment of Inertia of Magic Squares, Mathematical Gazette, Vol. 88, No. 511 (March 2004), 151-153, JSTOR:3621372.
- 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).
- Index entries for sequences related to moment of inertia.
Programs
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Mathematica
Array[(#^2*(#^4 - 1))/12 &, 31, 2] (* or *) Drop[CoefficientList[Series[-5 x^2*(x + 1) (x^2 + 4 x + 1)/(x - 1)^7, {x, 0, 32}], x], 2] (* Michael De Vlieger, Apr 13 2021 *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{5,60,340,1300,3885,9800,21840},40] (* Harvey P. Dale, Apr 03 2023 *) -
PARI
a(n) = (n^2 * (n^4 - 1))/12 \\ Felix Fröhlich, May 31 2021
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PARI
Vec(-5*x^2*(x+1)*(x^2+4*x+1)/(x-1)^7 + O(x^30)) \\ Felix Fröhlich, May 31 2021
Formula
a(n) = (n^2 * (n^4 - 1))/12.
G.f.: -5*x^2*(x+1)*(x^2+4*x+1) / (x-1)^7. - Colin Barker, Dec 10 2012
a(n) = Sum_{i=0..n^2-1} (k+i)^2 - (k*n + A027480(n-1))^2. - Charlie Marion, May 08 2021
From Amiram Eldar, Jul 03 2025: (Start)
Sum_{n>=2} 1/a(n) = 3*Pi*coth(Pi) - 2*Pi^2 + 21/2.
Sum_{n>=2} (-1)^n/a(n) = 3*Pi*cosech(Pi) + Pi^2 - 21/2. (End)
Extensions
More terms from Colin Barker, Dec 10 2012