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%I #33 Nov 19 2024 06:34:32
%S 12,66,228,714,2172,6546,19668,59034,177132,531426,1594308,4782954,
%T 14348892,43046706,129140148,387420474,1162261452,3486784386,
%U 10460353188,31381059594,94143178812,282429536466,847288609428,2541865828314,7625597484972,22876792454946
%N a(n) = 9*3^n - 15.
%C a(n) is the first Zagreb index of the Hanoi graph H[n].
%C The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph.
%C The M-polynomial of the Hanoi graph H[n] is M(H[n],x,y) = 6*x^2*y^3 + (3/2)*(3^n - 5)*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 I. Gutman and K. C. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
%H Eric. W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F O.g.f.: 6*x*(2 + 3*x)/((1 - x)*(1 - 3*x)).
%F E.g.f.: 3*(1 - exp(x))*(2 - 3*exp(x) - 3*exp(2*x)). - _Bruno Berselli_, Nov 14 2016
%F a(n) = 3*A168613(n+1). - _R. J. Mathar_, Apr 07 2022
%p seq(9*3^n-15, n = 1..30);
%t Table[9 3^n - 15, {n, 1, 30}] (* _Bruno Berselli_, Nov 14 2016 *)
%o (Magma) [9*3^n-15: n in [1..30]]; // _Bruno Berselli_, Nov 14 2016
%o (PARI) a(n)=3^(n+2)-15 \\ _Charles R Greathouse IV_, Nov 14 2016
%Y Cf. A277105.
%K nonn,easy
%O 1,1
%A _Emeric Deutsch_, Nov 05 2016