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

0,3

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

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).

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

Example

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

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 *)

Prog

(PARI) for(n=0, 50, print1(n^2*(n^2-9)/4", "))
(Magma)[n^2*(n^2+2*n-3)/4: n in [0..40]]; // Vincenzo Librandi, Sep 09 2011

Crossrefs

Cf. A000096, A000217, A173963.

Sequence in context: A137117 A064675 A135713 * A338996 A349919 A212783

Adjacent sequences: A085737 A085738 A085739 * A085741 A085742 A085743

Keyword

nonn,easy

Author

Jon Perry, Jul 21 2003

Status

approved