A085740 a(n) = T(n)^2 - n^2, where T(n) is a triangular number.

0, 0, 5, 27, 84, 200, 405, 735, 1232, 1944, 2925, 4235, 5940, 8112, 10829, 14175, 18240, 23120, 28917, 35739, 43700, 52920, 63525, 75647, 89424, 105000, 122525, 142155, 164052, 188384, 215325, 245055, 277760, 313632, 352869, 395675, 442260

Offset: 0

Author

Jon Perry, Jul 21 2003

Keywords

• nonn
• easy

Comments

a(n) is the dimension of the second Cartan power of sl(n, C), which is the irreducible representation of sl(n, C) the highest weight of which is twice that of the adjoint representation. - Daniel J. F. Fox, Jan 01 2006

Also the Harary index of the n X n rook graph. - Eric W. Weisstein, Jun 20 2017

a(n) is the dimension of the space of curvature tensors of Kähler type with vanishing Ricci trace on a Hermitian vector space of real dimension 2n. - Daniel J. F. Fox, Nov 21 2018

Examples

			a(3) = T(3)^2 - 3^2 = 6^2 - 9 = 36-9 = 27.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• M. Sitaramayya, Curvature tensors in Kaehler manifolds, Transactions of the AMS, 183 (September 1973), 341-353.
• Eric Weisstein's World of Mathematics, Harary Index
• Eric Weisstein's World of Mathematics, Rook Graph
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. A000096, A000217, A173963.

Programs

Magma

[n^2*(n^2+2*n-3)/4: n in [0..40]]; // Vincenzo Librandi, Sep 09 2011

Mathematica

Table[(n - 1) n^2 (n + 3)/4, {n, 20}] (* Eric W. Weisstein, Jun 20 2017 *)
Table[PolygonalNumber[n]^2 - n^2, {n, 20}] (* Eric W. Weisstein, Jun 20 2017 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 27, 84, 200}, 20] (* Eric W. Weisstein, Jun 20 2017 *)
CoefficientList[Series[(x (-5 - 2 x + x^2))/(-1 + x)^5, {x, 0, 20}],
  x] (* Eric W. Weisstein, Jun 20 2017 *)

PARI

for(n=0,50,print1(n^2*(n^2-9)/4","))

Formula

a(n) = n^2*(n^2+2*n-3)/4.

a(n) = A000096(n)*A000217(n-1).

a(n) = 4*A173963(n+1). - Reinhard Zumkeller, Mar 03 2010

G.f.: x^2*(5+2*x-x^2)/(1-x)^5. - Colin Barker, Mar 17 2012