%I #25 Sep 08 2022 08:44:58
%S 0,1,3,14,40,90,175,308,504,780,1155,1650,2288,3094,4095,5320,6800,
%T 8568,10659,13110,15960,19250,23023,27324,32200,37700,43875,50778,
%U 58464,66990,76415,86800,98208,110704,124355,139230,155400,172938
%N Number of scalars which can be constructed from the Riemann tensor and metric tensor in n dimensions.
%H G. C. Greubel, <a href="/A050297/b050297.txt">Table of n, a(n) for n = 1..5000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannTensor.html">Riemann Tensor</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(2) = 1, otherwise a(n) = n*(n-1)*(n-2)*(n+3)/12 = A005701(n-3).
%F From _Chai Wah Wu_, Aug 31 2016: (Start)
%F 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 G.f.: x^2*(x^5 - 5*x^4 + 10*x^3 - 9*x^2 + 2*x - 1)/(x - 1)^5. (End)
%t 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 Join[{0, 1}, Table[n (n - 1) (n - 2) (n + 3) / 12, {n, 3, 40}]] (* _Vincenzo Librandi_, May 13 2017 *)
%o (PARI) 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 (Magma) [0,1] cat [n*(n-1)*(n-2)*(n+3)/12: n in [3..60]]; // _Vincenzo Librandi_, May 13 2017
%Y Cf. A005701.
%K nonn,easy
%O 1,3
%A _Eric W. Weisstein_