A277108 a(n) = 4*n*(n+5).

24, 56, 96, 144, 200, 264, 336, 416, 504, 600, 704, 816, 936, 1064, 1200, 1344, 1496, 1656, 1824, 2000, 2184, 2376, 2576, 2784, 3000, 3224, 3456, 3696, 3944, 4200, 4464, 4736, 5016, 5304, 5600, 5904, 6216, 6536, 6864, 7200, 7544, 7896, 8256, 8624, 9000, 9384, 9776

Offset: 1

Author

Emeric Deutsch, Nov 05 2016

Keywords

• nonn
• easy

Comments

a(n) is the second Zagreb index of the helm graph H[n] (n>=3).

The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.

The M-polynomial of the Helm graph H[n] is M(H[n]; x,y) = n*x*y^4 + n*x^4*y^4 + n*x^4*y^n. - Emeric Deutsch, May 11 2018

The helm graph H[n] is the graph obtained from an n-wheel graph by adjoining a pendant edge at each node of the cycle. - Emeric Deutsch, May 11 2018

a(n) - 16*n + 1 is a square. - Muniru A Asiru, Jun 01 2018

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..5000
• Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
• Eric Weisstein's World of Mathematics, Helm Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A028557, A055998, A132761.

Programs

GAP

List([1..50],n->4*n*(n+5)); # Muniru A Asiru, Jun 01 2018

Maple

seq(4*n^2+20*n, n = 1 .. 40);

Mathematica

Table[4 n (n + 5), {n, 40}] (* or *)
Rest@ CoefficientList[Series[8 x (3 - 2 x)/(1 - x)^3, {x, 0, 40}], x] (* Michael De Vlieger, Nov 06 2016 *)

PARI

a(n)=4*n*(n+5) \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: 8*z*(3-2*z)/(1-z)^3.

a(n) = 4*A028557(n) = 8*A055998(n).

From Elmo R. Oliveira, Jan 28 2025: (Start)

E.g.f.: 4*exp(x)*x*(6 + x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)