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%I #43 Sep 30 2023 13:07:38
%S 5,60,340,1300,3885,9800,21840,44280,83325,147620,248820,402220,
%T 627445,949200,1398080,2011440,2834325,3920460,5333300,7147140,
%U 9448285,12336280,15925200,20345000,25742925,32284980,40157460,49568540,60749925,73958560,89478400
%N Moment of inertia of all magic squares of order n.
%H Michael De Vlieger, <a href="/A126275/b126275.txt">Table of n, a(n) for n = 2..10000</a>
%H Peter Loly, <a href="https://web.archive.org/web/20220723032820/http://home.cc.umanitoba.ca/~loly/MathGaz.pdf">The Invariance of the Moment of Inertia of Magic Squares</a>, Mathematical Gazette, Vol. 88, No. 511 (March 2004), 151-153, JSTOR:<a href="https://www.jstor.org/stable/3621372">3621372</a>.
%H Ivars Peterson, <a href="https://www.sciencenews.org/article/magic-square-physics">Magic Square Physics</a>, Science News online, Jul 01, 2006; Vol. 170, No. 1.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%H <a href="/index/Mo#moment_of_inertia">Index entries for sequences related to moment of inertia</a>.
%F a(n) = (n^2 * (n^4 - 1))/12.
%F G.f.: -5*x^2*(x+1)*(x^2+4*x+1) / (x-1)^7. - _Colin Barker_, Dec 10 2012
%F a(n) = Sum_{i=0..n^2-1} (k+i)^2 - (k*n + A027480(n-1))^2. - _Charlie Marion_, May 08 2021
%t Array[(#^2*(#^4 - 1))/12 &, 31, 2] (* or *)
%t 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 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{5,60,340,1300,3885,9800,21840},40] (* _Harvey P. Dale_, Apr 03 2023 *)
%o (PARI) a(n) = (n^2 * (n^4 - 1))/12 \\ _Felix Fröhlich_, May 31 2021
%o (PARI) Vec(-5*x^2*(x+1)*(x^2+4*x+1)/(x-1)^7 + O(x^30)) \\ _Felix Fröhlich_, May 31 2021
%Y Cf. A027480.
%K easy,nonn
%O 2,1
%A _Jonathan Vos Post_, Dec 23 2006
%E More terms from _Colin Barker_, Dec 10 2012
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