login
A213820
Principal diagonal of the convolution array A213819.
6
2, 18, 60, 140, 270, 462, 728, 1080, 1530, 2090, 2772, 3588, 4550, 5670, 6960, 8432, 10098, 11970, 14060, 16380, 18942, 21758, 24840, 28200, 31850, 35802, 40068, 44660, 49590, 54870, 60512, 66528, 72930, 79730, 86940, 94572, 102638, 111150, 120120, 129560, 139482
OFFSET
1,1
COMMENTS
Every term is even: a(n) = 2*A002414(n).
a(n) is the first Zagreb index of the graph obtained by joining one vertex of a complete graph K[n] with each vertex of a second complete graph K[n]. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. - Emeric Deutsch, Nov 07 2016
LINKS
Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
FORMULA
a(n) = -n + n^2 + 2*n^3.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: f(x)/g(x), where f(x) = 2*x*(1 + 5*x) and g(x) = (1-x)^4.
From Amiram Eldar, Mar 12 2023: (Start)
Sum_{n>=1} 1/a(n) = (4*log(2) - 1)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi - 4*log(2) + 1)/3. (End)
MATHEMATICA
(See A213819.)
a[n_] := 2*n^3 + n^2 - n; Array[a, 50] (* Amiram Eldar, Mar 12 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Jul 04 2012
STATUS
approved