A121145 Numbers of isomers of unbranched a-4-catapolyoctagons - see Brunvoll reference for precise definition.

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

Author

N. J. A. Sloane, Aug 13 2006

Keywords

• nonn

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 Brunvoll 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).

Programs

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

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)

Extensions

More terms from Petros Hadjicostas, Jul 24 2019 using the "master formula" in the reference