A304381 - OEIS

%I #18 May 09 2020 07:22:40

%S 32,168,412,764,1224,1792,2468,3252,4144,5144,6252,7468,8792,10224,

%T 11764,13412,15168,17032,19004,21084,23272,25568,27972,30484,33104,

%U 35832,38668,41612,44664,47824,51092,54468,57952,61544,65244,69052,72968,76992,81124,85364

%N a(n) = 54*n^2 - 26*n + 4 (n>=1).

%C a(n) is the second Zagreb index of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference.

%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 O(n,n) is M(O(n,n); x,y) = 4*(n+1)x^2*y^2 + 8(n-1)x^2 *y^3 + (6n^2 - 10n+4)x^3*y^3.

%C More generally, the M-polynomial of O(m,n) is M(O(m,n); x,y) =2(m+n+2)x^2*y^2+4(m+n-2)x^2 *y^3+(6mn-5m-5n+4)x^3*y^3.

%H Colin Barker, <a href="/A304381/b304381.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. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, <a href="https://joam.inoe.ro/articles/computing-topological-indices-of-certain-networks/">Computing topological indices of certain networks</a>, J. of Optoelectronics and Advanced Materials, 18, No. 9-10, 2016, 884-892.

%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 13 2018: (Start)

%F G.f.: 4*x*(8 + 18*x + 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(54*n^2-26*n+4, n = 1 .. 40);

%t Table[54n^2-26n+4,{n,40}] (* or *) LinearRecurrence[{3,-3,1},{32,168,412},40] (* _Harvey P. Dale_, Mar 21 2020 *)

%o (PARI) Vec(4*x*(8 + 18*x + x^2) / (1 - x)^3 + O(x^40)) \\ _Colin Barker_, May 13 2018

%Y Cf. A304380.

%K nonn,easy

%O 1,1

%A _Emeric Deutsch_, May 13 2018