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a(n) = 27*n^2/2 + 45*n/2 - 12 (n>=1).
2

%I #17 Dec 30 2023 13:32:26

%S 24,87,177,294,438,609,807,1032,1284,1563,1869,2202,2562,2949,3363,

%T 3804,4272,4767,5289,5838,6414,7017,7647,8304,8988,9699,10437,11202,

%U 11994,12813,13659,14532,15432,16359,17313,18294,19302,20337,21399,22488,23604,24747,25917,27114,28338,29589,30867,32172,33504

%N a(n) = 27*n^2/2 + 45*n/2 - 12 (n>=1).

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

%C The second Zagreb index of a simple connected graph is the sum of the degree products 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="/A304375/b304375.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.: 3*x*(8 + 5*x - 4*x^2) / (1 - x)^3.

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

%F (End)

%p seq(27*n^2/2 + 45*n/2 - 12, n=1..50);

%t Table[(27n^2)/2+(45n)/2-12,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{24,87,177},50] (* _Harvey P. Dale_, Dec 30 2023 *)

%o (PARI) a(n) = 27*n^2/2 + 45*n/2 - 12; \\ _Altug Alkan_, May 12 2018

%o (PARI) Vec(3*x*(8 + 5*x - 4*x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 12 2018

%Y Cf. A304374.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 12 2018