A304837 - OEIS

%I #16 May 23 2018 11:39:51

%S 0,348,1668,3960,7224,11460,16668,22848,30000,38124,47220,57288,68328,

%T 80340,93324,107280,122208,138108,154980,172824,191640,211428,232188,

%U 253920,276624,300300,324948,350568,377160,404724,433260,462768,493248,524700,557124,590520,624888,660228,696540,733824,772080,811308

%N a(n) = 6*(n - 1)*(81*n - 104) for n >= 1.

%C a(n) is the first Zagreb index of the hex derived network HDN1(n) from the Manuel et al. reference (see HDN1(4) in Fig. 8).

%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 HDN1(n) is M(HDN1(n); x, y) = 12*x^3*y^5 + (18*(n-2))*x^3*y^7 + (6*(3*n^2-9*n+7))*x^3*y^12 + 12*x^5*y^7 + 6*x^5*y^12 + (6*(n-3))*x^7*y^7 + (12*(n-2))*x^7*y^12 + (3*(n-2)*(3*n-5)*x^12*y^12.

%C 54*a(n) + 529 is a square. - _Bruno Berselli_, May 22 2018

%H Colin Barker, <a href="/A304837/b304837.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 P. Manuel, R. Bharati, I. Rajasingh, and Chris Monica M, <a href="https://doi.org/10.1016/j.jda.2006.09.002">On minimum metric dimension of honeycomb networks</a>, J. Discrete Algorithms, 6, 2008, 20-27.

%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*(29 + 52*x)/(1 - x)^3. - _Bruno Berselli_, May 22 2018

%p seq(624-1110*n+486*n^2, n = 1 .. 45);

%t Table[6 (n - 1) (81 n - 104), {n, 1, 50}] (* _Bruno Berselli_, May 22 2018 *)

%o (GAP) List([1..50], n->486*n^2-1110*n+624); # _Muniru A Asiru_, May 22 2018

%o (PARI) concat(0, Vec(12*x^2*(29 + 52*x)/(1 - x)^3 + O(x^40))) \\ _Colin Barker_, May 23 2018

%Y Cf. A082040, A304836, A304838.

%K nonn,easy

%O 1,2

%A _Emeric Deutsch_, May 21 2018