OFFSET
2,3
COMMENTS
Number of structural isomers of alkenes C_n H_{2n} with n carbon atoms.
Number of unicyclic graphs of n nodes where a double-edge replaces the cycle, [A217781], end-points of the double-edge of out-degrees <= 2, other nodes having out-degrees <= 3.
Number of rooted trees on n+1 nodes where the root has degree 2, the 2 children of the root have out-degrees <= 2, and the other nodes have out-degrees <= 3.
See illustration of initial terms. - Washington Bomfim, Nov 30 2020
REFERENCES
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..500
Washington Bomfim, Illustration of initial terms
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
H. R. Henze and C. M. Blair, The number of structural isomers of the more important types of aliphatic compounds, J. Amer. Chem. Soc., 56 (1) (1934), 157-157.
C.-W. Lam, A Mathematical Relationship between the Number of Isomers of Alkenes and Alkynes: A Result Established from the Enumeration of Isomers of Alkenes from Alky Biradicals, J. Math. Chem., 23, 421 (1998).
R. J. Mathar, Illustration for graphs up to 6 carbons
R. C. Read, Some recent results in chemical enumeration, Lect. Notes Math. 303 (1972), 243-259.
R. C. Read, Some recent results in chemical enumeration, Preprint, circa 1972. (Annotated scanned copy)
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 28.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
FORMULA
a(n) = b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b(n/2)(b(n/2) + 1)/2 when n is even or b(1)b(n-1) + b(2)b(n-2) + b(3)b(n-3) + ... + b((n-1)/2)b((n + 1)/2) when n is odd, where b(n) = A000642(n). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
a(n) = Sum_{k=1..(n-1)/2}( f(k) * f(n-k) ) + [n mod 2 = 0] * ( f(n/2)^2 + f(n/2) ) / 2 where f(n) = A000642(n+1). - Washington Bomfim, Nov 29 2020
G.f.: (g(x^2) + g(x)^2)/2 where x*g(x) is the g.f. of A000642. - Andrew Howroyd, Dec 01 2020
PROG
seq(n)={my(g=G(n), h=(subst(g, x, x^2) + g^2)/2); Vec(subst(h, x, x^2) + h^2)/2} \\ Andrew Howroyd, Dec 01 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 24 2008
STATUS
approved