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A181617 Molecular topological indices of the complete graph K_n. 7
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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
KEYWORD
nonn,easy,changed
AUTHOR
Eric W. Weisstein, Jul 10 2011
EXTENSIONS
More terms from Joerg Arndt, Mar 24 2014
STATUS
approved

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Last modified February 23 06:27 EST 2024. Contains 370267 sequences. (Running on oeis4.)