A181617 Molecular topological indices of the complete graph K_n.

0, 4, 24, 72, 160, 300, 504, 784, 1152, 1620, 2200, 2904, 3744, 4732, 5880, 7200, 8704, 10404, 12312, 14440, 16800, 19404, 22264, 25392, 28800, 32500, 36504, 40824, 45472, 50460, 55800, 61504, 67584, 74052, 80920, 88200, 95904, 104044, 112632, 121680, 131200

Offset: 1

Author

Eric W. Weisstein, Jul 10 2011

Keywords

• nonn
• easy

Comments

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

For n > 3, also the detour index of the (n-1)-helm graph. - Eric W. Weisstein, Dec 16 2017

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
• Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
• Eric Weisstein's World of Mathematics, Complete Graph.
• Eric Weisstein's World of Mathematics, Detour Index.
• Eric Weisstein's World of Mathematics, Helm Graph.
• Eric Weisstein's World of Mathematics, Molecular Topological Index.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002411.

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

Programs

Magma

[2*n*(n-1)^2: n in [1..50]]; // Vincenzo Librandi, Mar 24 2014

Mathematica

CoefficientList[Series[4 x (1 + 2 x)/(1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 24 2014 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 24, 72}, 50] (* Harvey P. Dale, Jun 16 2016 *)
Table[2 n (n - 1)^2, {n, 20}] (* Eric W. Weisstein, Dec 16 2017 *)

PARI

a(n) = 2*n*(n-1)^2; \\ Joerg Arndt, Mar 24 2014

Formula

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

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

G.f.: 4*x^2*(1+2*x)/(1-x)^4. - Colin Barker, Nov 04 2012

From Amiram Eldar, Jan 22 2023: (Start)

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

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

Extensions

More terms from Joerg Arndt, Mar 24 2014