A213820 Principal diagonal of the convolution array A213819.

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

Clark Kimberling, Table of n, a(n) for n = 1..1000

Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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

Cf. A002414, A213819, A328910.

Sequence in context: A114109 A085293 A119118 * A078837 A232155 A368755

Adjacent sequences: A213817 A213818 A213819 * A213821 A213822 A213823

Keyword

nonn,easy

Author

Clark Kimberling, Jul 04 2012

Status

approved