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A121145
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Numbers of isomers of unbranched a-4-catapolyoctagons - see Brunvoll reference for precise definition.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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)
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LINKS
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FORMULA
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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)
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MAPLE
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f:= n -> (1/4) * (n + (n + 8)*5^(n-3) + (1 - (-1)^n)*5^(floor(n/2) - 1)):
f(1):= 1:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Petros Hadjicostas, Jul 24 2019 using the "master formula" in the reference
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STATUS
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approved
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