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A245828 Szeged index of the grid graph P_n X P_n.

%I A245828 #31 Sep 08 2022 08:46:09
%S A245828 0,16,216,1280,5000,15120,38416,86016,174960,330000,585640,988416,
%T A245828 1599416,2497040,3780000,5570560,8018016,11302416,15638520,21280000,
%U A245828 28523880,37715216,49252016,63590400,81250000,102819600,128963016,160425216,198038680,242730000,295526720,357564416,430094016,514489360
%N A245828 Szeged index of the grid graph P_n X P_n.
%H A245828 Vincenzo Librandi, <a href="/A245828/b245828.txt">Table of n, a(n) for n = 1..1000</a>
%H A245828 S. Klavzar, A. Rajapakse and I. Gutman, <a href="http://dx.doi.org/10.1016/0893-9659(96)00071-7">The Szeged and the Wiener index of graphs</a>, Appl. Math. Lett., Vol. 9, No. 5 (1996), pp. 45-49.
%H A245828 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A245828 a(n) = (1/3)*n^4*(n^2 - 1).
%F A245828 G.f.: 8*x^2*(1+x)*(2+11*x+2*x^2)/(1-x)^7.
%F A245828 a(n) = A245826(n,n).
%F A245828 a(n) = 4 * A208954(n).
%F A245828 From _Amiram Eldar_, Jan 09 2022: (Start)
%F A245828 Sum_{n>=2} 1/a(n) = 33/4 - Pi^2/2 - Pi^4/30.
%F A245828 Sum_{n>=2} (-1)^n/a(n) = 7*Pi^4/240 + Pi^2/4 - 21/4. (End)
%p A245828 a := proc (n) options operator, arrow: (1/3)*n^4*(n^2-1) end proc: seq(a(n), n = 1 .. 40);
%t A245828 CoefficientList[Series[8 x (1 + x) (2 + 11 x + 2 x^2)/(1 - x)^7, {x, 0, 40}], x] (* _Vincenzo Librandi_, Aug 07 2014 *)
%t A245828 Table[(n^4 (n^2-1))/3,{n,40}] (* _Harvey P. Dale_, Mar 25 2021 *)
%o A245828 (Magma) [n^4*(n^2-1)/3: n in [1..40]]; // _Vincenzo Librandi_, Aug 07 2014
%Y A245828 Cf. A245826, A208954, A245827.
%K A245828 nonn,easy
%O A245828 1,2
%A A245828 _Emeric Deutsch_, Aug 06 2014