OFFSET
2,4
COMMENTS
From Petros Hadjicostas, May 26 2019: (Start)
Let I(r, k) be the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes, which are generated from catafusenes by converting k of its r hexagons to tetragons. According to Cyvin et al. (1996), for r >= k, we have I(r, k) = 1/4 *(binomial(r, k) + (r - 2)! * (r^2 + (4 * k - 1) * r + 4 * k * (k - 2)) * 3^(r - k - 2)/(k! * (r - k)!) + (2 + (-1)^k - (-1)^r) * (binomial(floor(r/2), floor(k/2)) + 2 * binomial(floor(r/2) - 1, floor(k/2) - 1)) * 3^(floor(r/2) - floor(k/2) - 1)). See Eq. (48) on p. 503 in the paper.
Letting k = 0 - 10, we get the eleven columns of Table 2 on p. 501 of Cyvin et al. (1996). (We need r >= max(k, 2) because the number of hexagons r should be greater than or equal to the number of converted polygons k.)
(End)
LINKS
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of Molecular structure 376 (Issues 1-3) (1996), 495-505. See Table 2 on p. 501.
FORMULA
For the element T(n, k) in row n >= 2 and column k >= 0 (such that max(k, 2) <= n), we have T(n, k) = I(r = n, k), where I(r, k) is given above in the comments. - Petros Hadjicostas, May 26 2019
EXAMPLE
Triangle begins (rows start at n = 2 and columns at k = 0):
1, 1, 1;
2, 3, 3, 1;
4, 7, 9, 3, 1;
10, 23, 29, 16, 5, 1;
25, 69, 99, 62, 27, 5, 1;
70, 229, 351, 275, 132, 39, 7, 1;
196, 731, 1249, 1121, 643, 221, 55, 7, 1;
574, 2385, 4437, 4584, 2997, 1278, 367, 72, 9, 1;
1681, 7657, 15597, 18012, 13458, 6678, 2322, 540, 93, 9, 1;
...
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Feb 09 2019
EXTENSIONS
Name edited by Petros Hadjicostas, May 26 2019
STATUS
approved