cp's OEIS Frontend

From Petros Hadjicostas, May 25 2019: (Start)

According to the Schoenflies notation used in Chemistry, the point group C_{2v} is the cyclic group C_2 "with the addition of" 2 "mirror planes containing the axis of rotation (vertical planes)" (see the Wikipedia article about the Schoenflies notation).

According to the Wikipedia article about Molecular symmetry, a simple description of typical geometry of this group is "angular (H2O) or see-saw (SF4)" (where SF4 = sulfur tetrafluoride).

According to Cyvin et al. (1996, p. 496), a catafusene is a "simply connected catacondensed polyhex". "It is a system consisting of congruent regular hexagons, where any two hexagons either share exactly one edge or are disjointed". "A tri-4-catafusene is generated by contraction of exactly three of its hexagons to tetragons" (see p. 499).

In general (see p. 496), "any alpha-q-catafusene (q = 3-5) can be generated from a catafusene on converting alpha of its hexagons to q-gons by contraction". Here, obviously, alpha = 3 and q = 4.

Here, a(n) is the number of (non-equivalent) unbranched tri-4-catafusenes under the symmetry point group C_{2v}, where n = number of polygons (also known as "rings") in the unbranched tri-4-catafusenes. (Hence, n - 3 is the total number of the remaining hexagons since 3 of the original hexagons became tetragons.) In the paper, the letter r is used to denote the number of polygons rather than the letter n (see p. 496).

The numbers (a(n): n >= 3) = (a(r): r >= 3) appear in Table 1 on p. 500 of Cyvin et al. (1996) under the point group C_{2v}.

In the paper, a(n) is denoted by M_r (p. 500) and is called the total number of "mirror-symmetrical (C_{2v}) systems" (here r = n). It is decomposed into the sum C_r + L_r, where C_r is the number of "centrosymmetrical (C_{2h}) systems" (p. 499) and L_r = (1/2)*binomial(r, 3) - (1/8)*(1-(-1)^r)*(r-1) (see Eq. (15) on p. 499). The numbers L_r count some kind of symmetry described on p. 499 of the paper, but it is unclear what they exactly count.

The point group C_{2h} mentioned above (with the centrosymmetrical systems) is the cyclic group C_2 "with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (horizontal plane)" (see the Wikipedia articles about Molecular symmetry and the Schoenflies notation).

Unfortunately, the letter C is used for two different purposes in the paper: to denote various point groups in three dimensions and to denote the number of non-equivalent systems of contracted catafusenes under some kind of symmetry.

The numbers (C_r: r >= 0), which apparently appear in Table 1 (p. 500) under the point group C_{2h} (see above), satisfy C_{2*i} = 0 and C_{2*i+3} = (1/2)*Sum_{1 <= j <= i} j*a_{i+1,j} (see Eq. (17), p. 500), where the double array a_{i,j} appears on p. 498 (Section 5) of the paper. We have a_{i,j} = binomial(i-1, j-1)*2^(i-j) = A038207(i-1, j-1) for i, j >= 1 (see Eq. (32), p. 502 in the paper).

Putting the above information together, we can easily prove the formulas for a(n) that appear below in the FORMULA section.

Note that the notation for the current sequence a(n) should not be confused with the double array a_{i,j} defined on p. 498 of the paper.

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See the comments of sequences A323939, A323941, and A323942 for explanations. - Petros Hadjicostas, May 26 2019

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.)

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