A305154 - OEIS

%I #23 Sep 08 2022 08:46:21

%S 45,81,153,297,585,1161,2313,4617,9225,18441,36873,73737,147465,

%T 294921,589833,1179657,2359305,4718601,9437193,18874377,37748745,

%U 75497481,150994953,301989897,603979785,1207959561,2415919113,4831838217,9663676425,19327352841,38654705673,77309411337,154618822665,309237645321

%N a(n) = 36*2^n + 9.

%C a(n) is the second Zagreb index of the dendrimer D[n], defined pictorially in Fig. 1 of the Heydari 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 D[n] is M(D[n]; x, y) = 3*2^n*x*y^3 + 6*x^2*y^3 + 3*(2^n - 1)*x^3*y^3 (n>=0).

%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. Heydari and I. Gutman, <a href="http://www.pmf.kg.ac.rs/kjs/volumes/kjs32/kjs32heydarigutman57.pdf">On the terminal Wiener index of thorn graphs</a>, Kragujevac J. Sci., 32, 2010, 57-64.

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

%F From _Vincenzo Librandi_, May 28 2018: (Start)

%F G.f.: 9*(5 - 6*x)/((1 - 2*x)*(1 - x)).

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

%F a(n) = 9*A000051(n+2). - _R. J. Mathar_, Jul 22 2022

%p seq(36*2^n + 9, n = 0..40);

%t Table[36 2^n + 9, {n, 0, 33}] (* _Vincenzo Librandi_, May 28 2018 *)

%t LinearRecurrence[{3,-2},{45,81},40] (* _Harvey P. Dale_, Jan 08 2020 *)

%o (Magma) [36*2^n+9: n in [0..33]]; // _Vincenzo Librandi_, May 28 2018

%Y Cf. A305153.

%K nonn,easy

%O 0,1

%A _Emeric Deutsch_, May 27 2018