A121101 Catapolyoctagons (see Cyvin et al. for precise definition).

1, 1, 3, 9, 39, 169, 819, 3969, 19719, 97969, 489219, 2442969, 12211719, 61042969, 305199219, 1525917969, 7629511719, 38147167969, 190735449219, 953675292969, 4768374511719, 23841862792969, 119209304199219, 596046472167969, 2980232312011719, 14901161315917969, 74505806335449219

Offset: 1

Author

N. J. A. Sloane, Aug 11 2006

Keywords

• nonn

Comments

From Petros Hadjicostas, Jul 24 2019: (Start)

The sequence (a(n): n >= 1) counts the isomers of unbranched alpha-4-catapoly-q-qons with alpha = 0 and q = 8. It appears in Table 21 (p. 12) in Brunvoll et al. (1997).

An unbranched alpha-4-catapoly-q-gon consists of alpha tetragons and n - alpha q-gons (where q > 4). Thus, n is the total number of polygons in the unbranched catacondensed polygonal system. Since we have alpha = 0 and q = 8 for this sequence, n counts the octagons.

The formula for a(n) below follows from the "master formula" I_{ra} in Exhibit 4 (p. 13) in Brunvoll et al. (1997) with alpha = 0 and q = 8 provided that a binomial coefficient of the form binomial(k, s) with s < 0 is set to zero.

Amazingly, the empirical g.f. of Colin Barker below is correct and follows easily from the formula for a(n) given below (with a(1) = 1).

(End)

References

• S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134(1) (1997), 55-70; see Table I (p. 58).

Links

• J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems representing polyclic conjugated hydrocarbons: unbalanced catacondensed systems with tetragons and q-gons, J. Molec. Struct. (Theochem), 364 (1996), 1-13.

Crossrefs

Cf. A121102, A121177.

Programs

Magma

[1] cat [(1/4)*(1+5^(n-2)+2*(2-(-1)^n)*5^((n div 2)-1)): n in [2..30]]; // Vincenzo Librandi, Jul 26 2019

Mathematica

Join[{1}, Table[(1/4) (1 + 5^(r - 2) + 2 (2 - (-1)^r) 5^(Floor[r/2] - 1)), {r, 2, 30}]] (* Vincenzo Librandi, Jul 26 2019 *)

Sage

def A121101_list(prec):
    P. = PowerSeriesRing(ZZ, default_prec=prec)
    def g(x): return x*(10*x^4-21*x^3+3*x^2+5*x-1)/((x-1)*(5*x-1)*(5*x^2-1))
    return P(g(x)).list()
print(A121101_list(27)) # Peter Luschny, Jul 26 2019

Formula

G.f.: x*(10*x^4 - 21*x^3 + 3*x^2 + 5*x - 1) / ((x - 1)*(5*x - 1)*(5*x^2 - 1)). - Colin Barker, Aug 29 2013

a(r) = (1/4) * (1 + 5^(r-2) + 2 * (2-(-1)^r) * 5^(floor(r/2) - 1)) for r >= 2. - Petros Hadjicostas, Jul 24 2019

Extensions

More terms from Petros Hadjicostas, Jul 24 2019 using the "master formula" in the references.