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a(n) = 225*2^n - 235.
1

%I #22 Nov 19 2024 23:46:41

%S -10,215,665,1565,3365,6965,14165,28565,57365,114965,230165,460565,

%T 921365,1842965,3686165,7372565,14745365,29490965,58982165,117964565,

%U 235929365,471858965,943718165,1887436565,3774873365,7549746965,15099494165,30198988565,60397977365,120795954965

%N a(n) = 225*2^n - 235.

%C a(n) is the second Zagreb index of the Wang's helicene-based nanostar DNS[n]. 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. The pictorial definition of DNS[n] can be viewed in the H. Shabani A. R. et al. reference (it is denoted DNS_{2}[n]).

%C The M-polynomial of the Wang's helicene-based dendrimer DNS[n] is M(DNS[n],x,y) = (2*2^n - 1)*x*y^3 + (6*2^n -4)*x^2*y^2 + (10*2^n - 12)*x^2*y^3 + (15*2^n - 16)*x^3*y^3.

%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 H. Shabani, A. R. Ashrafi, and I. Gutman, <a href="http://studia.ubbcluj.ro/arhiva/abstract_en.php?editie=CHEMIA&amp;nr=4&amp;an=2010&amp;id_art=8624">Geometric-arithmetic index: an algebraic approach</a>, Studia UBB, Chemia, 55, No. 4, 107-112, 2010.

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

%F G.f.: 5*(-2 + 49*x)/((1 - x)*(1 - 2*x)).

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

%p seq(225*2^n-235, n = 0..35)

%t Table[225*2^n - 235, {n, 0, 12}] (* or *)

%t CoefficientList[Series[5 (49 x - 2)/((1 - x) (1 - 2 x)), {x, 0, 12}], x] (* _Michael De Vlieger_, Nov 14 2016 *)

%t LinearRecurrence[{3,-2},{-10,215},30] (* _Harvey P. Dale_, May 28 2020 *)

%Y Cf. A278124.

%K sign,easy,changed

%O 0,1

%A _Emeric Deutsch_, Nov 13 2016