OFFSET
3,2
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).
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 09 2019
EXTENSIONS
Name edited by Petros Hadjicostas, May 26 2019
More terms using equation (48) in the paper from Petros Hadjicostas, May 26 2019
STATUS
approved