A323942 Irregular triangle read by rows giving the total number of isomers (nonisomorphic systems) of unbranched k-4-catafusenes.

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

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

Table of n, a(n) for n=2..64.

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

Column k = 0 is A001998. Column k = 3 is A323941.

Sequence in context: A274604 A089283 A288531 * A323944 A077941 A077990

Adjacent sequences: A323939 A323940 A323941 * A323943 A323944 A323945

Keyword

nonn,tabf

Author

N. J. A. Sloane, Feb 09 2019

Extensions

Name edited by Petros Hadjicostas, May 26 2019

Status

approved