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A228607
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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.
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2
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OFFSET
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1,1
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REFERENCES
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R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 161-162.
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LINKS
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FORMULA
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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.
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).
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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