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 A121101 Catapolyoctagons (see Cyvin et al. for precise definition). 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 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)  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. = 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: A149026 A149027 A180741 * A280066 A287063 A080635 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

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Last modified October 20 15:29 EDT 2019. Contains 328267 sequences. (Running on oeis4.)