A304374 - OEIS

%I #17 Apr 11 2024 15:50:06

%S 24,72,138,222,324,444,582,738,912,1104,1314,1542,1788,2052,2334,2634,

%T 2952,3288,3642,4014,4404,4812,5238,5682,6144,6624,7122,7638,8172,

%U 8724,9294,9882,10488,11112,11754,12414,13092,13788,14502,15234,15984,16752,17538,18342,19164,20004,20862,21738,22632,23544

%N a(n) = 9*n^2 + 21*n - 6 (n>=1).

%C a(n) is the first Zagreb index of the triangular benzenoid T(n) (see the M. Ghorbani et al. references).

%C The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.

%C The M-polynomial of the triangular benzenoid T(n) is M(T(n); x,y) = 6*x^2*y^2 + 6*(n-1)*x^2*y^3 + 3*n*(n-1)*x^3*y^3/2.

%H Colin Barker, <a href="/A304374/b304374.txt">Table of n, a(n) for n = 1..1000</a>

%H E. Deutsch and Sandi Klavzar, <a href="http://dx.doi.org/10.22052/ijmc.2015.10106">M-polynomial and degree-based topological indices</a>, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.

%H M. Ghorbani and M. Ghazi, <a href="http://www.chalcogen.ro/837_Ghorbani.pdf">Computing geometric arithmetic index of some graphs</a>, Digest J. of Nanomaterials and Biostructures, 5, No. 4, 2010, 837-841.

%H M. Ghorbani and M. Ghazi, <a href="http://www.chalcogen.ro/1107_Ghorbani.pdf">Computing some topological indices of triangular benzenoid</a>, Digest J. of Nanomaterials and Biostructures, 5, No. 4, 2010, 1107-1111.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Colin Barker_, May 12 2018: (Start)

%F G.f.: 6*x*(2 - x)*(2 + x) /(1 - x)^3.

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

%F (End)

%p seq(9*n^2 + 21*n - 6, n=1..50);

%t Table[9n^2+21n-6,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{24,72,138},50] (* _Harvey P. Dale_, Apr 11 2024 *)

%o (PARI) Vec(6*x*(2 - x)*(2 + x) /(1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 12 2018

%o (PARI) a(n) = 9*n^2 + 21*n - 6; \\ _Altug Alkan_, May 12 2018

%Y Cf. A304375.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 12 2018