A277984 - OEIS

%I #19 Sep 08 2022 08:46:17

%S 0,24,156,396,744,1200,1764,2436,3216,4104,5100,6204,7416,8736,10164,

%T 11700,13344,15096,16956,18924,21000,23184,25476,27876,30384,33000,

%U 35724,38556,41496,44544,47700,50964,54336,57816,61404,65100,68904,72816,76836,80964,85200

%N a(n) = 6*n*(9*n-5).

%C For n>=1, a(n) is the first Zagreb index of the circumcoronene B[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 definition of the circumcoronene can be viewed in the Gutman et al. and in the Farahani et al. references.

%C The M-polynomial of the circumcoronene B[n] is M(B[n],x,y) = 6*x^2*y^2 + 12*(n-1)*x^2*y^3 + 3*(3*n-2)*(n-1)*x^3*y^3.

%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. R. Farahani, M. R. Rajesh Kanna, M. K. Jamil, and M. Imran, <a href="http://www.sci-int.com/pdf/3797860791%20a%201%203251-3255%20Mohammad%20R.%20Farahani-MATH%20-CO--WAQAS+++.pdf">Computing the M-polynomial of benzenoid molecular graphs</a>, Sci. Int. (Lahore), 28(4), 2016, 3251-3255.

%H I. Gutman, S. J. Cyvin, and V. Ivanov-Petrovic, <a href="https://www.degruyter.com/downloadpdf/j/zna.1998.53.issue-8/zna-1998-0810/zna-1998-0810.xml">Topological properties of circumcoronenes</a>, Z. Naturforsch., 53a, 1998, 699-703.

%H I. Gutman and K. C. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.

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

%F G.f.: 12*x*(2+7*x)/(1-x)^3.

%p seq(54*n^2-30*n, n = 0 .. 40);

%t CoefficientList[Series[12 x (2 + 7 x) / (1 - x)^3, {x, 0, 45}], x] (* _Vincenzo Librandi_, Nov 13 2016 *)

%t LinearRecurrence[{3,-3,1},{0,24,156},50] (* _Harvey P. Dale_, Apr 10 2022 *)

%o (Magma) [6*n*(9*n-5): n in [0..45]]; // _Vincenzo Librandi_, Nov 13 2016

%o (PARI) a(n)=6*n*(9*n-5) \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Cf. A277985.

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Nov 11 2016