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

1,3

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

Table of n, a(n) for n=1..27.

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.

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

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 *)

Prog

(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
(Sage)
def A121101_list(prec):
    P.<x> = 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

Crossrefs

Cf. A121102, A121177.

Sequence in context: A149027 A330795 A180741 * A280066 A360876 A287063

Adjacent sequences: A121098 A121099 A121100 * A121102 A121103 A121104

Keyword

nonn

Author

N. J. A. Sloane, Aug 11 2006

Extensions

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

Status

approved