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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

References

• 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.

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from Emeric Deutsch, Mar 14 2004