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Total number of isomers (nonisomorphic systems) of unbranched tri-4-catafusenes as a function of the number of hexagons (see Cyvin et al. (1996) for precise definition).
4

%I #42 Apr 25 2023 09:21:53

%S 1,3,16,62,275,1121,4584,18012,69573,262495,974704,3562714,12859127,

%T 45881213,162093320,567579192,1971791241,6801382203,23309839120,

%U 79421200630,269160513115,907726206233,3047449980392,10188384020372,33930772031565,112595241877911,372383348102640,1227721195083922

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

%C From _Petros Hadjicostas_, May 26 2019: (Start)

%C 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.

%C 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.)

%C (End)

%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 the last column of Table 1 on p. 500.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (16,-102,304,-247,-1056,3372,-3168,-2223,8208,-8262,3888,-729).

%F a(n) = I(r = n, k = 3) in the formula above in the comments (for n >= 3). - _Petros Hadjicostas_, May 26 2019

%F G.f.: -x^3*(-1 +13*x -70*x^2 +192*x^3 -250*x^4 +22*x^5 +402*x^6 -672*x^7 +663*x^8 -387*x^9 +72*x^10) / ( (-1+3*x^2)^2 *(3*x-1)^4 *(x-1)^4 ). - _R. J. Mathar_, Jul 25 2019

%p CyvinI := proc(r,k)

%p if r >= k then

%p 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));

%p else

%p -1;

%p end if;

%p end proc:

%p A323941 := proc(n)

%p CyvinI(n,3) ;

%p end proc:

%p seq(A323941(n),n=3..30) ; # _R. J. Mathar_, Jul 25 2019

%t CyvinI[r_, k_] := If[r >= k, 1/4 * (Binomial[r, k] + (r-2)! * (r^2 + (4k - 1) * r + 4k * (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)), -1];

%t a[n_] := CyvinI[n, 3];

%t Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Apr 25 2023 *)

%Y Cf. A323939, A323940.

%K nonn

%O 3,2

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

%E Name edited by _Petros Hadjicostas_, May 26 2019

%E More terms using equation (48) in the paper from _Petros Hadjicostas_, May 26 2019