A228607 - OEIS

%I #29 Nov 26 2025 15:59:40

%S 54,9450,286679250,263199759084281250,

%T 221721055245240563933498289550781250,

%U 157320497971930517299046640166039915248640240419548633694648742675781250

%N a(n) is the number of independent vertex subsets (i.e., the Merrifield-Simmons index) of the triangulane T[n] defined in the Khalifeh et al. and Deutsch et al. references.

%D R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.

%H Emeric Deutsch, <a href="/A228607/b228607.txt">Table of n, a(n) for n = 1..9</a>

%H E. Deutsch, S. Klavzar, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match70/n2/match70n2_627-644.pdf">Computing the Hosoya polynomial of graphs from primary subgraphs</a>, MATCH Commun. Math. Comput. Chem., 70, 2013, 627-644.

%H M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, <a href="http://nopr.niscair.res.in/handle/123456789/2215">Computing Wiener and Kirchhoff indices of a triangulane</a>, Indian J. Chemistry, 47A, 2008, 1503-1507.

%H H. Prodinger and R. F. Tichy, <a href="https://www.fq.math.ca/Scanned/20-1/prodinger.pdf">Fibonacci numbers of graphs</a>, Fibonacci Quarterly, 20, 1982, 16-21.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Triangulane">Triangulane</a>

%F a(n) = d(n)^3 + 3*c(n)*d(n)^2, where c(1) = 1, d(1) = 3, c(n) = d(n-1)^2, d(n) = 2*c(n-1)*d(n-1) + d(n-1)^2 for n>=2.

%F If we replace the initial conditions for c and d by c[1] = x and d[1] = 1 + 2x, respectively, and the first equation by c[n] = x*d[n-1]^2, then a(n) will yield the independence polynomial of the triangulane T(n). For example, for n=2 one finds 1 + 21x + 180x^2 + 816x^3 + 2112x^4 + 3120x^5 + 2432x^6 + 768x^7 (checked with the Maple Graph Theory package).

%F d(n) = A338293(n+1). - _R. J. Mathar_, Jul 22 2022

%p c[1] := 1: d[1] := 3: for n from 2 to 10 do c[n] := d[n-1]^2; d[n] := 2*c[n-1]*d[n-1]+d[n-1]^2 end do: seq(d[n]^3+3*c[n]*d[n]^2, n = 1 .. 7);

%K nonn

%O 1,1

%A _Emeric Deutsch_, Dec 17 2013