A277979 a(n) = 4*n^2 + 18*n.

0, 22, 52, 90, 136, 190, 252, 322, 400, 486, 580, 682, 792, 910, 1036, 1170, 1312, 1462, 1620, 1786, 1960, 2142, 2332, 2530, 2736, 2950, 3172, 3402, 3640, 3886, 4140, 4402, 4672, 4950, 5236, 5530, 5832, 6142, 6460, 6786, 7120, 7462, 7812, 8170, 8536, 8910, 9292, 9682

Offset: 0

Author

Emeric Deutsch, Nov 08 2016

Keywords

• nonn
• easy

Comments

For n>=3, a(n) is the first Zagreb index of the double-wheel graph DW[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.

The double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.

The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y)= 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.

4*a(n) + 81 is a square. - Bruno Berselli, May 08 2018

Examples

			a(3) = 90. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the first Zagreb index is 6*6 + 6*9 = 90.
		

Links

• E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A139576, A277980.

Subsequence of A028569.

Programs

Magma

[4*n^2+18*n: n in [0..50]]; // Vincenzo Librandi, Nov 09 2016

Maple

seq(4*n^2+18*n,n = 0..50);

Mathematica

Table[4 n^2 + 18 n, {n, 0, 50}] (* Vincenzo Librandi, Nov 09 2016 *)
LinearRecurrence[{3,-3,1},{0,22,52},50] (* Harvey P. Dale, Mar 01 2022 *)

PARI

a(n)=4*n^2+18*n \\ Charles R Greathouse IV, Jun 17 2017

Formula

O.g.f.: 2*x*(11 - 7*x)/(1 - x)^3.

E.g.f.: 2*x*(11 + 2*x)*exp(x).

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

a(n) = 2*A139576(n).