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a(n) = 5*(n+1)*(9*n+4).
2

%I #40 Nov 15 2024 18:21:32

%S 20,130,330,620,1000,1470,2030,2680,3420,4250,5170,6180,7280,8470,

%T 9750,11120,12580,14130,15770,17500,19320,21230,23230,25320,27500,

%U 29770,32130,34580,37120,39750,42470,45280,48180,51170,54250,57420,60680,64030,67470,71000,74620

%N a(n) = 5*(n+1)*(9*n+4).

%C The first Zagreb index of the single-defect 5-gonal nanocone CNC(5,n) (see definition in the Doslic et al. reference, p. 27).

%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 CNC(5,n) is M(CNC(5,n); x,y) = 5*x^2*y^2 + 10*n*x^2*y^3 + 5*n*(3*n+1)*x^3*y^3/2.

%C More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.

%H Colin Barker, <a href="/A304507/b304507.txt">Table of n, a(n) for n = 0..1000</a>

%H Emeric 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, Vol. 6, No. 2, 2015, pp. 93-102.

%H T. Doslic and M. Saheli, <a href="http://dx.doi.org/10.22061/jmns.2011.460">Augmented eccentric connectivity index of single-defect nanocones</a>, Journal of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.

%H A. Khaksar, M. Ghorbani, and H. R. Maimani, <a href="https://oam-rc.inoe.ro/download.php?idu=1353">On atom bond connectivity and GA indices of nanocones</a>, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.

%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 14 2018: (Start)

%F G.f.: 10*(2 + 7*x)/(1 - x)^3.

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

%F a(n) = 10*A062708(n+1) for n >= 0. - _Robert G. Wilson v_, May 14 2018

%F a(n) = 5*A011862(9*n+7) = 5*A108579(6*n+7). - _Bruno Berselli_, May 15 2018

%F From _Elmo R. Oliveira_, Nov 15 2024: (Start)

%F E.g.f.: 5*exp(x)*(4 + 22*x + 9*x^2).

%F a(n) = 5*A017209(n)*A008587(n+1). (End)

%p seq((5*(n+1))*(9*n+4), n = 0 .. 40);

%t Array[5 (# + 1) (9 # + 4) &, 41, 0] (* or *)

%t LinearRecurrence[{3, -3, 1}, {20, 130, 330}, 41] (* or *)

%t CoefficientList[Series[10 (2 + 7 x)/(1 - x)^3, {x, 0, 40}], x] (* _Michael De Vlieger_, May 14 2018 *)

%o (PARI) a(n) = 5*(n+1)*(9*n+4); \\ _Altug Alkan_, May 14 2018

%o (PARI) Vec(10*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 14 2018

%o (GAP) List([0..50], n -> 5*(n+1)*(9*n+4)); # _Muniru A Asiru_, May 15 2018

%Y Cf. A008587, A011862, A017209, A062708, A108579, A304508.

%K nonn,easy,changed

%O 0,1

%A _Emeric Deutsch_, May 14 2018