A323940 - OEIS

%I #42 Nov 07 2019 07:01:46

%S 0,1,8,52,244,1093,4490,17952,69304,262385,973916,3562532,12856716,

%T 45880933,162085694,567578784,1971766704,6801381633,23309759728,

%U 79421199860,269160256356,907726205221,3047449152562,10188384019072,33930769372904

%N Number of nonisomorphic systems (isomers) for the unsymmetrical schemes (group C_s) for unbranched tri-4-catafusenes as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).

%C See the comments of sequences A323939, A323941, and A323942 for explanations. - _Petros Hadjicostas_, May 26 2019

%H Georg Fischer, <a href="/A323940/b323940.txt">Table of n, a(n) for n = 3..103</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 on 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_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-71,116,259,-1246,1013,2520,-5187,594,5931,-4428, -1215,2430,-729).

%F a(n) = (1/8) * (1 - (-1)^n) * ((n - 1) - (n + 3) * 3^((n - 5)/2)) + (1/8) * (n^2 + 11 * n + 12) * (n - 2) * 3^(n - 6) - (1/4) * binomial(n, 3) for n >= 3. - _Petros Hadjicostas_, May 26 2019

%F G.f.: x^4*(1 -6*x +11*x^2 -32*x^3 +182*x^4 -346*x^5 -122*x^6 +950*x^7 -831*x^8 +336*x^9 -297*x^10 +90*x^11) / ( (1+x)^2*(3*x^2-1)^2*(3*x-1)^4*(x-1)^4 ). - _R. J. Mathar_, Jul 25 2019

%p # Calculates a(r) = AA(r), where r = n is the number of hexagons.

%p # Crude numbers:

%p JJ := proc(i) sum(binomial(j + 1, 3)*binomial(i - 2, j - 1)*2^(i - 1 - j), j = 1 .. i - 1); end proc;

%p # Linearly annelated systems of D_{2h} symmetry:

%p DD := proc(r) 1/4*(1 - (-1)^r)*(r - 1); end proc;

%p # Linearly annelated systems of C_{2v} symmetry:

%p LL := proc(r) 1/2*binomial(r, 3) - (1/8 - 1/8*(-1)^r)*(r - 1); end proc;

%p # Centrosymmetrical (C_{2h}) systems:

%p CC := proc(n) 1/24*(1 - (-1)^n)*((3 + n)*3^(1/2*n - 3/2) - 3*n + 3); end proc;

%p # Total mirror-symmetrical (C_{2v}) systems:

%p MM := proc(n) CC(n) + LL(n); end proc;

%p # Unsymmetrical (C_s) systems:

%p AA := proc(r) 1/4*(JJ(r) - DD(r) - 2*CC(r) - 2*MM(r)); end proc;

%p # Generate sequence:

%p for m from 3 to 100 do AA(m); end do; # _Petros Hadjicostas_, May 26 2019

%t LinearRecurrence[{14, -71, 116, 259, -1246, 1013, 2520, -5187, 594, 5931, -4428, -1215, 2430, -729}, {0, 1, 8, 52, 244, 1093, 4490, 17952, 69304, 262385, 973916, 3562532, 12856716, 45880933}, 100] (* from the g.f., _Georg Fischer_, Nov 07 2019 *)

%Y Cf. A323939, A323941, A323942.

%K nonn,easy

%O 3,3

%A _N. J. A. Sloane_, Feb 09 2019

%E Name edited by _Petros Hadjicostas_, May 26 2019

%E More terms using various equations in Cyvin et al. (1996) from _Petros Hadjicostas_, May 26 2019