%I #69 Feb 19 2024 01:36:00

%S 0,4,24,72,160,300,504,784,1152,1620,2200,2904,3744,4732,5880,7200,

%T 8704,10404,12312,14440,16800,19404,22264,25392,28800,32500,36504,

%U 40824,45472,50460,55800,61504,67584,74052,80920,88200,95904,104044,112632,121680,131200

%N Molecular topological indices of the complete graph K_n.

%C a(n) = the area of a trapezoid with vertices at (n-1,n), (n,n-1), ((n-1)^2,n^2), and (n^2,(n-1)^2). - _J. M. Bergot_, Mar 23 2014

%C For n > 3, also the detour index of the (n-1)-helm graph. - _Eric W. Weisstein_, Dec 16 2017

%C a(n-3) is the maximum sigma irregularity over all maximal 2-degenerate graphs with n vertices. The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices). (The sigma irregularity of a graph is the sum of the squares of the differences between the degrees over all edges of the graph.) - _Allan Bickle_, Jun 14 2023

%H Vincenzo Librandi, <a href="/A181617/b181617.txt">Table of n, a(n) for n = 1..1000</a>

%H Allan Bickle and Zhongyuan Che, <a href="https://doi.org/10.1016/j.dam.2023.01.020">Irregularities of Maximal k-degenerate Graphs</a>, Discrete Applied Math. 331 (2023) 70-87.

%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteGraph.html">Complete Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DetourIndex.html">Detour Index</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HelmGraph.html">Helm Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MolecularTopologicalIndex.html">Molecular Topological Index</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 2*n*(n-1)^2.

%F a(n) = 4*A002411(n).

%F G.f.: 4*x^2*(1+2*x)/(1-x)^4. - _Colin Barker_, Nov 04 2012

%F From _Amiram Eldar_, Jan 22 2023: (Start)

%F Sum_{n>=2} 1/a(n) = Pi^2/12 - 1/2.

%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/24 - log(2) + 1/2. (End)

%t CoefficientList[Series[4 x (1 + 2 x)/(1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)

%t LinearRecurrence[{4, -6, 4, -1}, {0, 4, 24, 72}, 50] (* _Harvey P. Dale_, Jun 16 2016 *)

%t Table[2 n (n - 1)^2, {n, 20}] (* _Eric W. Weisstein_, Dec 16 2017 *)

%o (PARI) a(n) = 2*n*(n-1)^2; \\ _Joerg Arndt_, Mar 24 2014

%o (Magma) [2*n*(n-1)^2: n in [1..50]]; // _Vincenzo Librandi_, Mar 24 2014

%Y Cf. A002411.

%Y Cf. A011379, A181617, A270205 (sigma irregularities of maximal k-degenerate graphs).

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Jul 10 2011

%E More terms from _Joerg Arndt_, Mar 24 2014