This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A121145 #31 May 08 2025 08:05:24
%S A121145 1,1,4,16,85,439,2358,12502,66471,351565,1855784,9765628,51271097,
%T A121145 268554691,1403816410,7324218754,38147011723,198364257817,
%U A121145 1029968457036,5340576171880,27656556152349,143051147460943,739097600097662,3814697265625006,19669532800292975
%N A121145 Numbers of isomers of unbranched a-4-catapolyoctagons - see Brunvoll reference for precise definition.
%C A121145 From _Petros Hadjicostas_, Jul 24 2019: (Start)
%C A121145 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).
%C A121145 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.
%C A121145 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.
%C A121145 (End)
%H A121145 Robert Israel, <a href="/A121145/b121145.txt">Table of n, a(n) for n = 1..1428</a>
%H A121145 J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="https://doi.org/10.1016/0166-1280(95)04463-9">Isomer enumeration of polygonal systems representing polyclic conjugated hydrocarbons: unbalanced catacondensed systems with tetragons and q-gons</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13.
%H A121145 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (12,-41,0,205,-300,125).
%F A121145 From _Petros Hadjicostas_, Jul 24 2019: (Start)
%F A121145 a(n) = (1/4) * (n + (n + 8)*5^(n-3) + (1 - (-1)^n)*5^(floor(n/2) - 1)) for n >= 2.
%F A121145 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 ).
%F A121145 (End)
%p A121145 f:= n -> (1/4) * (n + (n + 8)*5^(n-3) + (1 - (-1)^n)*5^(floor(n/2) - 1)):
%p A121145 f(1):= 1:
%p A121145 map(f, [$1..40]); # _Robert Israel_, Jul 25 2019
%K A121145 nonn
%O A121145 1,3
%A A121145 _N. J. A. Sloane_, Aug 13 2006
%E A121145 More terms from _Petros Hadjicostas_, Jul 24 2019 using the "master formula" in the reference