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A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].

%I A242371 #70 Feb 16 2025 08:33:22
%S A242371 12,32,40,72,84,128,144,200,220,288,312,392,420,512,544,648,684,800,
%T A242371 840,968,1012,1152,1200,1352,1404,1568,1624,1800,1860,2048,2112,2312,
%U A242371 2380,2592,2664,2888,2964,3200,3280,3528,3612,3872,3960,4232,4324,4608,4704
%N A242371 Modified eccentric connectivity index of the cycle graph with n vertices, C[n].
%C A242371 The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph. This is a generalization of eccentric connectivity index.
%C A242371 a(n) = 4*A093353(n-1) = n*A168273(n) for n>2. - _Alois P. Heinz_, Jun 26 2014
%H A242371 Nilanjan De, <a href="/A242371/b242371.txt">Table of n, a(n) for n = 3..100</a>
%H A242371 N. De, S. M. A. Nayeem and A. Pal, <a href="http://camo.ici.ro/journal/vol16/v16a11.pdf">Bounds for modified eccentric connectivity index</a>, Advanced Modeling and Optimization, 16(1) (2014) 133-142.
%H A242371 N. De, S. M. A. Nayeem and A. Pal, <a href="http://arxiv.org/abs/1402.1870">Bounds for modified eccentric connectivity index</a>, arXiv:1402.1870 [math.CO], 2014.
%H A242371 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphEccentricity.html">Graph Eccentricity</a>
%F A242371 a(n) = 2*n*(n-1) if n is odd; and a(n) = 2*n^2 if n is even (n>2).
%F A242371 G.f.: -4*x^3*(3+5*x-4*x^2-2*x^3+2*x^4)/((x+1)^2*(x-1)^3). - _Alois P. Heinz_, Jun 26 2014
%e A242371 a(3) = 3*4 = 12 because there are 3 vertices and each vertex has eccentricity 1 and the total degree of neighboring vertices is 4.
%p A242371 a:= n-> n*(2*n-1+(-1)^n):
%p A242371 seq(a(n), n=3..60);  # _Alois P. Heinz_, Jun 26 2014
%t A242371 a[n_] := 2n(n-Boole[OddQ[n]]);
%t A242371 Table[a[n], {n, 3, 50}] (* _Jean-François Alcover_, Nov 28 2018 *)
%o A242371 (PARI) a(n) = if (n % 2, 2*n*(n-1), 2*n^2); \\ _Michel Marcus_, Jun 20 2014
%Y A242371 Cf. A093353, A168273, A206490, A238410, A238411.
%K A242371 nonn
%O A242371 3,1
%A A242371 _Nilanjan De_, Jun 08 2014