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

1,2

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).

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)

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 *)

Prog

(PARI) a(n) = 2*n*(n-1)^2; \\ Joerg Arndt, Mar 24 2014
(Magma) [2*n*(n-1)^2: n in [1..50]]; // Vincenzo Librandi, Mar 24 2014

Crossrefs

Cf. A002411.

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

Sequence in context: A212066 A336039 A364600 * A261256 A011915 A199904

Adjacent sequences: A181614 A181615 A181616 * A181618 A181619 A181620

Keyword

nonn,easy

Author

Eric W. Weisstein, Jul 10 2011

Extensions

More terms from Joerg Arndt, Mar 24 2014

Status

approved