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 = 1 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 = 1 and q = 8 for this sequence, n - 1 counts the octagons.
The formula for a(n) below follows from the "master formula" I_{ra} in Exhibit 4 (p. 13) in Brumvoll et al. (1997) with alpha = 1 and q = 8 provided that a binomial coefficient of the form binomial(k, s) with s < 0 is set to zero.
(End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..1428
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.
Index entries for linear recurrences with constant coefficients, signature (12,-41,0,205,-300,125).
FORMULA
From Petros Hadjicostas, Jul 24 2019: (Start)
a(n) = (1/4) * (n + (n + 8)*5^(n-3) + (1 - (-1)^n)*5^(floor(n/2) - 1)) for n >= 2.
G.f.: x - x^2*(1 -8*x +9*x^2 +57*x^3 -130*x^4 +55*x^5) /((-1+5*x^2) *(5*x-1)^2 *(x-1)^2 ).
(End)
MAPLE
f:= n -> (1/4) * (n + (n + 8)*5^(n-3) + (1 - (-1)^n)*5^(floor(n/2) - 1)):
f(1):= 1:
map(f, [$1..40]); # Robert Israel, Jul 25 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 13 2006
EXTENSIONS
More terms from Petros Hadjicostas, Jul 24 2019 using the "master formula" in the reference
STATUS
approved