A044045 -id:A044045 - cp's OEIS frontend

A121178 Four-column table read by rows: number of nonisomorphic systems of catafusenes in an example (see Cyvin et al. (1994) for precise definition).

1, 0, 0, 0, 2, 2, 0, 0, 6, 5, 1, 0, 19, 26, 5, 1, 71, 101, 31, 2, 274, 457, 160, 16, 1117, 1978, 825, 80, 4650, 8851, 4074, 473, 19819, 39481, 19902, 2517, 85710, 178043, 95920, 13431

Offset: 1

N. J. A. Sloane, Aug 15 2006

From Petros Hadjicostas, May 25 2019: (Start)
The sequence (a rectangular array) refers to Table 5 on p. 1179 of Cyvin et al. (1994). The zeroth row and the zeroth column from Table 5 do not appear in this sequence. The columns (alpha = 1, 2, 3, 4) refer to the number of appendages to the core.
The table refers to the example described on pp. 1177-1179 of the paper and especially to Figure 5 (p. 1178). The entries refer to the number of nonisomorphic systems of catafusenes in this example (as shown in Figure 5, p. 1178).
The rows of the table are indexed by the total number of hexagons in the appendages.
(End)

			The table begins:
     1,    0,    0,   0;
     2,    2,    0,   0;
     6,    5,    1,   0;
    19,   26,    5,   1;
    71,  101,   31,   2;
   274,  457,  160,  16;
  1117, 1978,  825,  80;
  4650, 8851, 4074, 473;
  ...

S. J. Cyvin, B. N. Cyvin, J. Brunvoll and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, Journal of Chemical Information and Modeling [formerly, J. Chem. Inform. Comput. Sci.], 34 (1994), pp. 1174-1180.
Eric Weisstein's World of Mathematics, Fusene.

Columns give A044045, A045903 - A045905. Row sums give A045906.

Corrected by N. J. A. Sloane, Apr 14 2013 (thanks to Michel Marcus for noticing that something was wrong)
Name edited by Petros Hadjicostas, May 25 2019
More terms from reference from Petros Hadjicostas, May 25 2019

A038392 Number of mono-4-polyhexes with n cells.

Original entry on oeis.org

1, 1, 2, 6, 19, 71, 274, 1117, 4650, 19819, 85710, 375712, 1664203, 7439593, 33515758, 152019560, 693625265, 3181528275, 14661581030, 67850297506, 315187646601, 1469195636293, 6869889703638, 32215399021901, 151467334017864, 713881817440421, 3372142139764434

Offset: 1

N. J. A. Sloane

J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298; see Eq. 16.

Robert Israel, Table of n, a(n) for n = 1..1437
S. J. Cyvin, Graph-theoretical studies on fluoranthenoids and fluorenoids. Part 1, Journal of Molecular Structure (Theochem), 262 (1992), 219-231.
N. Cyvin, E. Brendsdal, J. Brunvoll, S. J. Cyvin, A class of polygonal systems representing polycyclic conjugated hydrocarbons: Catacondensed monoheptafusenes, Monat. f. Chemie, 125 (1994), 1327-1337.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll and E. Brendsdal, Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes, Journal of Chemical Information and Modeling [formerly, J. Chem. Inform. Comput. Sci.], 34 (1994), pp. 1174-1180.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, Zhang Fuji, Guo Xiaofeng, and R. Tosic, Graph-theoretical studies on fluoranthenoids and fluorenoids: enumeration of some catacondensed systems, Journal of Molecular Structures (Theochem), 285 (1993), 179-185.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
Eric Weisstein's World of Mathematics, Fusene.

Apart from initial term, (A002212 + A007317)/2. See A044045 for another version.

f:= gfun:-rectoproc({(250*n^2-250*n)*a(n)+(-300*n^2-150*n)*a(n+1)+(-325*n^2-875*n-600)*a(n+2)+(475*n^2+2045*n+2100)*a(n+3)+(35*n^2+265*n+540)*a(n+4)+(-193*n^2-1691*n-3660)*a(n+5)+(49*n^2+563*n+1596)*a(n+6)+(17*n^2+211*n+648)*a(n+7)+(-9*n^2-135*n-504)*a(n+8)+(n^2+17*n+72)*a(n+9), a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 2, a(4) = 6, a(5) = 19, a(6) = 71, a(7) = 274, a(8) = 1117},a(n),remember):
map(f, [$1..50]); # Robert Israel, Oct 08 2017

f[z_] := Sqrt[5*z^2 - 6*z + 1]; g[z_] := (2*(1 - z^2) - (1-z)*f[z] - f[z^2])/ (4*(1-z)); Drop[ CoefficientList[ Series[ g[z], {z, 0, 24}], z], 1] (* Jean-François Alcover, Oct 13 2011, after Emeric Deutsch *)

G.f.: (2(1-z^2) - (1-z)f(z) - f(z^2))/(4(1-z)) where f(z) = sqrt(1-6z+5z^2). - Emeric Deutsch, Mar 14 2004

(250*n^2-250*n)*a(n)+(-300*n^2-150*n)*a(n+1)+(-325*n^2-875*n-600)*a(n+2)+(475*n^2+2045*n+2100)*a(n+3)+(35*n^2+265*n+540)*a(n+4)+(-193*n^2-1691*n-3660)*a(n+5)+(49*n^2+563*n+1596)*a(n+6)+(17*n^2+211*n+648)*a(n+7)+(-9*n^2-135*n-504)*a(n+8)+(n^2+17*n+72)*a(n+9) = 0. - Robert Israel, Oct 08 2017

More terms from Emeric Deutsch, Mar 14 2004

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A121178 Four-column table read by rows: number of nonisomorphic systems of catafusenes in an example (see Cyvin et al. (1994) for precise definition).

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A038392 Number of mono-4-polyhexes with n cells.

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A121070 Nonisomorphic catacondensed monoheptafusenes (see reference for precise definition).

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