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A050297 Number of scalars which can be constructed from the Riemann tensor and metric tensor in n dimensions.

%I A050297 #31 May 22 2025 08:33:55
%S A050297 0,1,3,14,40,90,175,308,504,780,1155,1650,2288,3094,4095,5320,6800,
%T A050297 8568,10659,13110,15960,19250,23023,27324,32200,37700,43875,50778,
%U A050297 58464,66990,76415,86800,98208,110704,124355,139230,155400,172938
%N A050297 Number of scalars which can be constructed from the Riemann tensor and metric tensor in n dimensions.
%H A050297 G. C. Greubel, <a href="/A050297/b050297.txt">Table of n, a(n) for n = 1..5000</a>
%H A050297 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RiemannTensor.html">Riemann Tensor</a>.
%H A050297 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A050297 a(2) = 1, otherwise a(n) = n*(n-1)*(n-2)*(n+3)/12 = A005701(n-3).
%F A050297 From _Chai Wah Wu_, Aug 31 2016: (Start)
%F A050297 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 7.
%F A050297 G.f.: x^2*(x^5 - 5*x^4 + 10*x^3 - 9*x^2 + 2*x - 1)/(x - 1)^5. (End)
%F A050297 From _Amiram Eldar_, May 22 2025: (Start)
%F A050297 Sum_{n>=2} 1/a(n) = 437/300.
%F A050297 Sum_{n>=2} (-1)^n/a(n) = 1547/300 - 32*log(2)/5. (End)
%t A050297 CoefficientList[Series[x^2*(x^5 - 5*x^4 + 10*x^3 - 9*x^2 + 2*x - 1)/(x - 1)^5, {x, 0, 50}], x] (* _G. C. Greubel_, May 12 2017 *)
%t A050297 Join[{0, 1}, Table[n (n - 1) (n - 2) (n + 3) / 12, {n, 3, 40}]] (* _Vincenzo Librandi_, May 13 2017 *)
%o A050297 (PARI) my(x='x+O('x^50)); concat([0], Vec(x^2*(x^5-5*x^4+10*x^3-9*x^2+2*x-1)/(x-1)^5)) \\ _G. C. Greubel_, May 12 2017
%o A050297 (Magma) [0,1] cat [n*(n-1)*(n-2)*(n+3)/12: n in [3..60]]; // _Vincenzo Librandi_, May 13 2017
%Y A050297 Cf. A005701.
%K A050297 nonn,easy
%O A050297 1,3
%A A050297 _Eric W. Weisstein_