cp's OEIS Frontend

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

It would be nice to have generating functions for any of A129012-A129021.

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

From Petros Hadjicostas, Nov 17 2019: (Start)

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).

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.

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 -> ...

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), ...

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

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.

(End)

This allows a single pentagonal ring among otherwise hexagonal rings.

The terms occur in groups of 5: X, X, Y, Z, Y, ..., except for the 0th term (corresponding to the C5H5 series). [Edited by Petros Hadjicostas, Nov 17 2019]

From Petros Hadjicostas, Nov 17 2019: (Start)

From all the papers listed in the Links, the author of this sequence considers Dias (1991) as the basic one for this sequence.

The starting compounds for each constant-isomer series are as follows (with the number of isomers inside parentheses): C5H5 (1), C9H7 (1), C12H8 (1), C15H9 (2), C18H10 (3), C23H11 (2), C26H12 (7), C31H13 (7), C36H14 (9), C41H15 (17), C48H16 (9), C53H17 (31), C60H18 (31), C67H19 (41), C74H20 (72), C83H21 (41), C90H22 (120), C99H23 (120), C108H24 (...), C117H25 (...), C128H26 (...), where "..." means that the corresponding number of isomers is not listed in the papers (probably because they are not known yet).

Starting with each one of the above compounds, the corresponding constant-isomer series is generated by the operator P(C_n H_s) -> C_{n + 2*s + 5} H_{s + 5} successively. For example, the first series is C5H5 -> C20H10 -> C45H15 -> C80H20 -> ...

The starting compound of each constant-isomer series is C_n H_s, where n = s + 2 + 2*floor((1/10) * (s^2 - 7*s + 6)), for s = 5, 7, 8, 9, 10, 11, 12, ... (i.e., we skip s = 6). Thus, a(0) = 1 corresponds to s = 5 (i.e., C5H5), and for m >= 1, a(m) corresponds to s = m + 6, i.e., to C_n H_s, where n = s + 2 + 2*floor((1/10) * (s^2 - 7*s + 6)). See Cyvin et al. (1993, p. 233). (End)