A304618 - OEIS

%I #29 May 22 2018 09:49:30

%S 90,402,930,1674,2634,3810,5202,6810,8634,10674,12930,15402,18090,

%T 20994,24114,27450,31002,34770,38754,42954,47370,52002,56850,61914,

%U 67194,72690,78402,84330,90474,96834,103410,110202,117210,124434,131874,139530,147402,155490,163794

%N a(n) = 108*n^2 - 228*n + 114 (n>=2).

%C For n>=2, a(n) is the first Zagreb index of the hexagonal network HX(n).

%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 hexagonal network HX(n) is M(HX(n); x,y) = 12*x^3*y^4 + 6*x^3*y^6 + 6*(n-3)*x^4*y^4 + 12*(n-2)*x^4*y^6 + (9*n^2-33*n+30)*x^6*y^6.

%C 3*a(n) + 19 is a square. - _Bruno Berselli_, May 18 2018

%H Colin Barker, <a href="/A304618/b304618.txt">Table of n, a(n) for n = 2..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 S. Hayat and M. Imran, <a href="https://doi.org/10.1016/j.amc.2014.04.091">Computation of topological indices of certain networks</a>, Applied Mathematics and Computation, 240, 2014, 213-228/

%H M. N. Husin and R. Hasni, <a href="https://doi.org/10.1504/IJNVO.2017.083543">More results on computation of topological indices of certain networks</a>, Int. J. Networking and Virtual Organisations, 17, No. 1, 2017, 46-63.

%H B. Rajan, A. William, C. Grigorius, and S. Stephen, <a href="http://www.compmath-journal.org/dnload/BHARATI-RAJAN-ALBERT-WILLIAM-CYRIAC-GRIGORIOUS-and-SUDEEP-STEPHEN/CMJV03I05P0530.pdf">On certain topological indices of silicate, honeycomb and hexagonal networks</a>, J. Comp. & Math. Sci., 3, No. 5, 2012, 530-535.

%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 18 2018: (Start)

%F G.f.: 6*x^2*(15 + 22*x - x^2) / (1 - x)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>4.

%F (End)

%p seq(114-228*n+108*n^2, n = 2 .. 40);

%o (GAP) List([2..40], n->108*n^2-228*n+114); # _Muniru A Asiru_, May 18 2018

%o (PARI) a(n) = 108*n^2 - 228*n + 114; \\ _Altug Alkan_, May 18 2018

%o (PARI) Vec(6*x^2*(15 + 22*x - x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 18 2018

%Y Cf. A304619.

%K nonn,easy

%O 2,1

%A _Emeric Deutsch_, May 17 2018