%I #87 Oct 24 2019 11:34:30
%S 0,2,5,10,22,28,65,60,172,110,461,182,1314,280,3977,408,12504,570,
%T 40021,770,128814,1012,414481,1300,1330052,1638,4253341,2030,13553978,
%U 2480,43049433,2992,136317872,3570,430471077,4218,1355976262,4940,4261630689,5740,13366013020,6622
%N Number of unbranched tri-4-catafusenes under the symmetry point group C_{2v} as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).
%C From _Petros Hadjicostas_, May 25 2019: (Start)
%C 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).
%C 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).
%C 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).
%C 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.
%C 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).
%C 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}.
%C 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.
%C 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).
%C 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.
%C 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).
%C Putting the above information together, we can easily prove the formulas for a(n) that appear below in the FORMULA section.
%C 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.
%C (End)
%H Colin Barker, <a href="/A323939/b323939.txt">Table of n, a(n) for n = 3..1000</a>
%H S. J. Cyvin, B. N. Cyvin and J. Brunvoll, <a href="https://dx.doi.org/10.1016/0022-2860(95)09039-8">Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons</a>, Journal of Molecular Structure 376 (Issues 1-3) (1996), 495-505. See Table 1 (p. 500).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Fusene.html">Fusene</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Molecular_symmetry">Molecular symmetry</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Point_groups_in_three_dimensions">Point groups in three dimensions</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyhex_(mathematics)">Polyhex (mathematics)</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schoenflies_notation">Schoenflies notation</a>.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,7,-16,-14,44,2,-48,15,18,-9).
%F From _Petros Hadjicostas_, May 26 2019: (Start)
%F a(n) = L(n) + C(n) for n >= 3, where L(n) = (1/2)*binomial(n, 3) - (1/8)*(1-(-1)^n)*(n-1) and C(n) = (1/24)*(1 - (-1)^n)*((n + 3)*3^((n-3)/2) - 3*(n-1)).
%F a(2*m) = binomial(2*m, 3)/2 = A006331(m - 1) for m >= 2, and a(2*m + 1) = binomial(2*m + 1, 3)/2 + (m + 2)*3^m/18 - m for m >= 1.
%F (End)
%F From _Colin Barker_, May 28 2019: (Start)
%F G.f.: x^4*(2 + x - 14*x^2 - x^3 + 22*x^4 - 3*x^5 + 2*x^6 - x^7) / ((1 - x)^4*(1 + x)^2*(1 - 3*x^2)^2).
%F a(n) = 2*a(n-1) + 7*a(n-2) - 16*a(n-3) - 14*a(n-4) + 44*a(n-5) + 2*a(n-6) - 48*a(n-7) + 15*a(n-8) + 18*a(n-9) - 9*a(n-10) for n>12.
%F (End)
%t a[n_] := (1/24) (3((-1)^n - 1) (n - 1) + 2 n (n - 1) (n - 2)+ ((-1)^(n + 1) + 1) (-3n + 3^((n - 3)/2) (n + 3) + 3));
%t a /@ Range[3, 44] (* _Jean-François Alcover_, Oct 24 2019 *)
%o (PARI) concat(0, Vec(x^4*(2 + x - 14*x^2 - x^3 + 22*x^4 - 3*x^5 + 2*x^6 - x^7) / ((1 - x)^4*(1 + x)^2*(1 - 3*x^2)^2) + O(x^40))) \\ _Colin Barker_, May 30 2019
%Y Cf. A006331, A038207, A323940, A323941.
%K nonn,easy
%O 3,2
%A _N. J. A. Sloane_, Feb 09 2019
%E Name edited by _Petros Hadjicostas_, May 26 2019
%E More terms from _Petros Hadjicostas_, May 26 2019