cp's OEIS Frontend

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

%I A121101 #37 Sep 08 2022 08:45:27
%S A121101 1,1,3,9,39,169,819,3969,19719,97969,489219,2442969,12211719,61042969,
%T A121101 305199219,1525917969,7629511719,38147167969,190735449219,
%U A121101 953675292969,4768374511719,23841862792969,119209304199219,596046472167969,2980232312011719,14901161315917969,74505806335449219
%N A121101 Catapolyoctagons (see Cyvin et al. for precise definition).
%C A121101 From _Petros Hadjicostas_, Jul 24 2019: (Start)
%C A121101 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).
%C A121101 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.
%C A121101 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.
%C A121101 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).
%C A121101 (End)
%D A121101 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).
%H A121101 J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="https://doi.org/10.1016/0166-1280(95)04463-9">Isomer enumeration of polygonal systems representing polyclic conjugated hydrocarbons: unbalanced catacondensed systems with tetragons and q-gons</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13.
%F A121101 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
%F A121101 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
%t A121101 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 *)
%o A121101 (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
%o A121101 (Sage)
%o A121101 def A121101_list(prec):
%o A121101     P.<x> = PowerSeriesRing(ZZ, default_prec=prec)
%o A121101     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))
%o A121101     return P(g(x)).list()
%o A121101 print(A121101_list(27)) # _Peter Luschny_, Jul 26 2019
%Y A121101 Cf. A121102, A121177.
%K A121101 nonn
%O A121101 1,3
%A A121101 _N. J. A. Sloane_, Aug 11 2006
%E A121101 More terms from _Petros Hadjicostas_, Jul 24 2019 using the "master formula" in the references.