This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085740 #36 Feb 16 2025 08:32:50
%S A085740 0,0,5,27,84,200,405,735,1232,1944,2925,4235,5940,8112,10829,14175,
%T A085740 18240,23120,28917,35739,43700,52920,63525,75647,89424,105000,122525,
%U A085740 142155,164052,188384,215325,245055,277760,313632,352869,395675,442260
%N A085740 a(n) = T(n)^2 - n^2, where T(n) is a triangular number.
%C A085740 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
%C A085740 Also the Harary index of the n X n rook graph. - _Eric W. Weisstein_, Jun 20 2017
%C A085740 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
%H A085740 Vincenzo Librandi, <a href="/A085740/b085740.txt">Table of n, a(n) for n = 0..10000</a>
%H A085740 M. Sitaramayya, <a href="https://www.ams.org/journals/tran/1973-183-00/S0002-9947-1973-0322722-1/S0002-9947-1973-0322722-1.pdf">Curvature tensors in Kaehler manifolds</a>, Transactions of the AMS, 183 (September 1973), 341-353.
%H A085740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HararyIndex.html">Harary Index</a>
%H A085740 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>
%H A085740 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F A085740 a(n) = n^2*(n^2+2*n-3)/4.
%F A085740 a(n) = A000096(n)*A000217(n-1).
%F A085740 a(n) = 4*A173963(n+1). - _Reinhard Zumkeller_, Mar 03 2010
%F A085740 G.f.: x^2*(5+2*x-x^2)/(1-x)^5. - _Colin Barker_, Mar 17 2012
%e A085740 a(3) = T(3)^2 - 3^2 = 6^2 - 9 = 36-9 = 27.
%t A085740 Table[(n - 1) n^2 (n + 3)/4, {n, 20}] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t A085740 Table[PolygonalNumber[n]^2 - n^2, {n, 20}] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t A085740 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 5, 27, 84, 200}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)
%t A085740 CoefficientList[Series[(x (-5 - 2 x + x^2))/(-1 + x)^5, {x, 0, 20}],
%t A085740 x] (* _Eric W. Weisstein_, Jun 20 2017 *)
%o A085740 (PARI) for(n=0,50,print1(n^2*(n^2-9)/4","))
%o A085740 (Magma)[n^2*(n^2+2*n-3)/4: n in [0..40]]; // _Vincenzo Librandi_, Sep 09 2011
%Y A085740 Cf. A000096, A000217, A173963.
%K A085740 nonn,easy
%O A085740 0,3
%A A085740 _Jon Perry_, Jul 21 2003