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A364922 a(n) is the square of the side length of a simplex whose n-dimensional inner hypervolume is equal to its (n-1)-dimensional surface hypervolume. As a result, the sequence starts at n=2.

%I A364922 #70 Aug 27 2024 13:53:59
%S A364922 48,216,640,1500,3024,5488,9216,14580,22000,31944,44928,61516,82320,
%T A364922 108000,139264,176868,221616,274360,336000,407484,489808,584016,
%U A364922 691200,812500,949104,1102248,1273216,1463340,1674000,1906624,2162688,2443716,2751280,3087000
%N A364922 a(n) is the square of the side length of a simplex whose n-dimensional inner hypervolume is equal to its (n-1)-dimensional surface hypervolume. As a result, the sequence starts at n=2.
%C A364922 Setting the generalized hypervolume formula equal to the surface hypervolume formula and solving for the side length x (and ignoring the x = 0 solution, as it would correspond to a simplex consisting of only a single point) gives x = sqrt(2*(n^3)*(n+1)).
%H A364922 Harvey P. Dale, <a href="/A364922/b364922.txt">Table of n, a(n) for n = 2..1000</a>
%H A364922 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A364922 a(n) = 2*n^3*(n+1) = 2*A179824(n+1).
%F A364922 From _Stefano Spezia_, Apr 13 2024: (Start)
%F A364922 G.f.: 4*x^2*(12 - 6*x + 10*x^2 - 5*x^3 + x^4)/(1 - x)^5.
%F A364922 a(n) = 4*A019582(n+1). (End)
%t A364922 Table[2*n^3*(n + 1), {n, 2, 50}] (* _Paolo Xausa_, Apr 18 2024 *)
%t A364922 LinearRecurrence[{5,-10,10,-5,1},{48,216,640,1500,3024},40] (* _Harvey P. Dale_, Aug 27 2024 *)
%o A364922 (Python)
%o A364922 def a(n): return 2 * n**3 * (n + 1)
%o A364922 print([a(n) for n in range(2, 50)])
%Y A364922 Cf. A019582, A179824.
%K A364922 easy,nonn
%O A364922 2,1
%A A364922 _Matt Moir_, Apr 13 2024