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A121145
Numbers of isomers of unbranched a-4-catapolyoctagons - see Brunvoll reference for precise definition.
2
1, 1, 4, 16, 85, 439, 2358, 12502, 66471, 351565, 1855784, 9765628, 51271097, 268554691, 1403816410, 7324218754, 38147011723, 198364257817, 1029968457036, 5340576171880, 27656556152349, 143051147460943, 739097600097662, 3814697265625006, 19669532800292975
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)
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
Sequence in context: A090013 A333370 A125793 * A144882 A298012 A006681
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