A323941 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).

1, 3, 16, 62, 275, 1121, 4584, 18012, 69573, 262495, 974704, 3562714, 12859127, 45881213, 162093320, 567579192, 1971791241, 6801382203, 23309839120, 79421200630, 269160513115, 907726206233, 3047449980392, 10188384020372, 33930772031565, 112595241877911, 372383348102640, 1227721195083922

Offset: 3

Author

N. J. A. Sloane, Feb 09 2019

Keywords

• nonn

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

• 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 the last column of Table 1 on p. 500.
• Index entries for linear recurrences with constant coefficients, signature (16,-102,304,-247,-1056,3372,-3168,-2223,8208,-8262,3888,-729).

Crossrefs

Cf. A323939, A323940.

Programs

Maple

CyvinI := proc(r,k)
    if r >= k then
        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));
    else
        -1;
    end if;
end proc:
A323941 := proc(n)
    CyvinI(n,3) ;
end proc:
seq(A323941(n),n=3..30) ; # R. J. Mathar, Jul 25 2019

Mathematica

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];
a[n_] := CyvinI[n, 3];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Apr 25 2023 *)

Formula

a(n) = I(r = n, k = 3) in the formula above in the comments (for n >= 3). - Petros Hadjicostas, May 26 2019

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

Extensions

Name edited by Petros Hadjicostas, May 26 2019

More terms using equation (48) in the paper from Petros Hadjicostas, May 26 2019