%I #32 Nov 18 2019 01:32:47

%S 1,1,1,2,4,4,12,19,19,46,70,70,162,239,239,504,726,726,1471,2062,2062

%N Isomer numbers for the constant-isomer series: the monradical, diradical, triradical, tetraradical, etc. series.

%C The terms occur in groups of three, X, X, Y.

%C From _Petros Hadjicostas_, Nov 17 2019: (Start)

%C When we count diradical isomers, the starting compounds for each constant-isomer series are as follows (with the number of diradical isomers inside parentheses): C22H12 (1), C30H14 (1), C40H16 (1), C50H18 (2), C62H20 (4), C76H22 (4), C90H24 (12), C106H26 (19), C124H28 (19), C142H30 (46), C162H32 (70), C184H34 (70), C206H36 (162), C230H38 (239), C256H40 (239), C282H42 (504), C310H44 (726), C340H46 (726), C370H48 (1471), C402H50 (2062), C436H52 (2062).

%C For some mysterious reason, the ground compounds of each series, which are listed above, obey the general formula C_{2*b(s)} H_{2*s}, where b(s) = A096777(s), for s = 6, 7, ..., 26.

%C Given a ground compound in a constant-isomer series of compounds, the series is determined by the operator P(C_n H_s) -> C_{n + 2*s + 6} H_{s + 6}. For example, the series corresponding to the C22H12 is C22H12 -> C52H18 -> C94H24 -> C148H30 -> C214H36 -> ...

%C As it can be seen in Dias (1996), the same numbers appear for the number of monoradical isomers for odd-carbon compounds starting with C13H9. See also Table 1 in Dias (1991, p. 128). Here we have the following starting compounds for each constant-isomer series (with the number of monoradical isomers in parentheses): C13H9 (1), C19H11 (1), C27H13(1), C35H15 (2), C45H17 (4), C57H19 (4), C69H21 (12), C83H23 (19), C99H25 (19), C115H27 (46), ...

%C For additional interpretations of these numbers (e.g., in terms of tetraradicals), see the equations and theory in Dias (1993).

%C The "base formulas for the smallest one-isomer polyradicals" appear in Section 7.6 in Dias (1996), which explains why we begin with C22H12 for the number of diradical isomers and with C13H9 for the number of monoradicals.

%C (End)

%D J. R. Dias, The Periodic Table Set as a Unifying Concept in Going from Benzenoid Hydrocarbons to Fullerene Carbons, in "The Periodic Table: Into the 21st Century", Edited by D. H. Rouvray and R. B. King, Research Studies Press Ltd, Baldock, Hertfordshire, England, 2004, 371-396.

%H Jerry Ray Dias, <a href="https://doi.org/10.1007/BF01126675">Constant-isomer benzenoid series and their polyradical subsets</a>, Theoretica Chimica Acta 81(3) (1991), 125-138; see Table 2 on p. 129.

%H Jerry Ray Dias, <a href="https://doi.org/10.1021/ci00011a017">Notes on constant-isomer series</a>, J. Chem. Inf. Comput. Sci. 33 (1993), pp. 117-127; see p. 123.

%H Jerry Ray Dias, <a href="https://doi.org/10.1016/0166-218X(95)00012-G">Graph theoretical invariants and elementary subgraphs of polyhex and polypent/polyhex systems of chemical relevance</a>, Discr. App. Math. 67 (1-3) (1996), pp. 79-114; see p. 98 (same numbers for the number of monoradical isomers for odd-carbon compounds starting with C13H9).

%Y Cf. A096777, A129012-A129021.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, based on email from Jerry R. Dias (DiasJ(AT)umkc.edu), May 08 2007