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

0, 1, 3, 14, 40, 90, 175, 308, 504, 780, 1155, 1650, 2288, 3094, 4095, 5320, 6800, 8568, 10659, 13110, 15960, 19250, 23023, 27324, 32200, 37700, 43875, 50778, 58464, 66990, 76415, 86800, 98208, 110704, 124355, 139230, 155400, 172938

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Eric Weisstein's World of Mathematics, Riemann Tensor.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005701.

Programs

Magma

[0,1] cat [n*(n-1)*(n-2)*(n+3)/12: n in [3..60]]; // Vincenzo Librandi, May 13 2017

Mathematica

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 *)
Join[{0, 1}, Table[n (n - 1) (n - 2) (n + 3) / 12, {n, 3, 40}]] (* Vincenzo Librandi, May 13 2017 *)

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

Formula

a(2) = 1, otherwise a(n) = n*(n-1)*(n-2)*(n+3)/12 = A005701(n-3).

From Chai Wah Wu, Aug 31 2016: (Start)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 7.

G.f.: x^2*(x^5 - 5*x^4 + 10*x^3 - 9*x^2 + 2*x - 1)/(x - 1)^5. (End)

From Amiram Eldar, May 22 2025: (Start)

Sum_{n>=2} 1/a(n) = 437/300.

Sum_{n>=2} (-1)^n/a(n) = 1547/300 - 32*log(2)/5. (End)