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%I #27 Jan 27 2024 18:05:24
%S 270,762,1746,3714,7650,15522,31266,62754,125730,251682,503586,
%T 1007394,2015010,4030242,8060706,16121634,32243490,64487202,128974626,
%U 257949474,515899170,1031798562,2063597346,4127194914,8254390050,16508780322,33017560866,66035121954,132070244130
%N a(n) = 492*2^n - 222.
%C a(n) is the first Zagreb index of the phenylacetylene dendrimer NSB[n] defined pictorially in the Yarahmadi references. 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 dendrimer NSB[n] is M(NSB[n],x,y) = 9*2^n*x*y^4 + (24*2^n - 12)*x^2*y^2 + (48*2^n - 24)*x^2*y^3 + (15*2^n - 9)*x^3*y^3 + 3*2^n*x^3*y^4.
%C From _Bruno Berselli_, Nov 16 2016: (Start)
%C In general, this type of formula b(n) = k*2^n - k (where n>=0 and h, k are given constants) has:
%C O.g.f.: (h - k - (h - 2*k)*x)/((1 - x)*(1 - 2*x));
%C E.g.f.: (-k + h*exp(x))*exp(x);
%C Linear recurrence: b(n) = 3*b(n-1) - 2*b(n-2);
%C Signature of the recurrence: (3,-2). (End)
%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 Z. Yarahmadi, <a href="http://dx.doi.org/10.22052/ijmc.2010.5154">Eccentric connectivity and augmented eccentric connectivity indices of N-branches phenylacetylenes nanostar dendrimers</a>, Iranian J. Math. Chem., 1, No. 2, 2010, 105-110.
%H Z. Yarahmadi and G. H. Fath-Tabar, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match65/n1/match65n1_201-208.pdf">The Wiener, Szeged, PI, Vertex PI, the first and second Zagreb indices of N-branched phenylacetylenes dendrimers</a>, MATCH: Commun. Math. Comput. Chem, 65 (2011) 201-208.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F O.g.f.: 6*(45 - 8*x)/((1 - x)*(1 - 2*x)).
%F E.g.f.: 6*(-37 + 82*exp(x))*exp(x).
%F a(n) = 3*a(n-1) - 2*a(n-2).
%p seq(492*2^n-222, n=0..35)
%t Table[492 2^n - 222, {n, 0, 35}] (* _Vincenzo Librandi_, Nov 16 2016 *)
%t LinearRecurrence[{3,-2},{270,762},30] (* _Harvey P. Dale_, Jan 27 2024 *)
%o (Magma) [492*2^n-222: n in [0..35]]; // _Vincenzo Librandi_, Nov 16 2016
%Y Cf. A278131.
%K nonn,easy
%O 0,1
%A _Emeric Deutsch_, Nov 15 2016
%E Edited by _Bruno Berselli_, Nov 16 2016
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