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a(n) = 136*2^n - 112.
3

%I #13 Jan 30 2023 18:35:05

%S 24,160,432,976,2064,4240,8592,17296,34704,69520,139152,278416,556944,

%T 1114000,2228112,4456336,8912784,17825680,35651472,71303056,142606224,

%U 285212560,570425232,1140850576,2281701264,4563402640,9126805392,18253610896,36507221904,73014443920,146028887952,292057776016,584115552144

%N a(n) = 136*2^n - 112.

%C a(n) is the first Zagreb index of the second type dendrimer nanostar NS2[n], defined pictorially in the Chen et al. reference (Fig. 1).

%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 NS2[n] is M(NS2[n];x,y) = 2*(4*2^n-1)*x^2*y^2 + 16*(2*n - 1)*x^2*y^3 + 4*(2^n - 1)*x^3*y^3 (n>=0).

%H S. Chen and J. Yang, <a href="http://www.m-hikari.com/imf-2011/5-8-2011/chenshuboIMF5-8-2011.pdf">Second-order and third-order connectivity indices of dendrimer nanostars</a>, International Mathematical Forum, 6, No, 5, 2011, 223-228.

%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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).

%F G.f.: 8*(3 + 11*x)/((1 - x)*(1 - 2*x)). - _Vincenzo Librandi_, May 27 2018

%p seq(136*2^n-112, n = 0 .. 40);

%t Table[136 2^n - 112, {n, 0, 33}] (* _Vincenzo Librandi_, May 27 2018 *)

%t LinearRecurrence[{3,-2},{24,160},40] (* _Harvey P. Dale_, Jan 30 2023 *)

%o (Magma) [136*2^n - 112: n in [0..33]]; // _Vincenzo Librandi_, May 27 2018

%Y Cf. A305163, A305164, A305166.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, May 27 2018