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 A125669 #14 May 25 2018 03:23:43
%S A125669 1,4,20,76,288,1005,3433,11324,36712,116809,367076,1140226,3510491,
%T A125669 10722708,32539364,98178211,294767639,881147521,2623934079,7787024985,
%U A125669 23039064263,67977412951,200072442611,587532484513,1721812143140,5036454320102,14706743476128
%N A125669 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 0. See the paper by Hendrickson and Parks for details.
%C A125669 Here 'alpha' is the number of atoms the two rings have in common.
%C A125669 Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles without any shared node and all nodes having degree at most 4. See A121162 for the special case of the two cycles having the same length. - _Andrew Howroyd_, May 25 2018
%H A125669 Andrew Howroyd, <a href="/A125669/b125669.txt">Table of n, a(n) for n = 6..200</a>
%H A125669 J. B. Hendrickson and C. A. Parks, <a href="https://doi.org/10.1021/ci00001a018">Generation and Enumeration of Carbon skeletons</a>, J. Chem. Inf. Comput. Sci., 31 (1991), pp. 101-107. See Table VII column 2 on page 104.
%e A125669 If n=6 then the number of bicyclics when 'alpha' = zero is 1.
%e A125669 If n=7 then the number of bicyclics when 'alpha' = zero is 4.
%e A125669 If n=8 then the number of bicyclics when 'alpha' = zero is 20.
%e A125669 If n=9 then the number of bicyclics when 'alpha' = zero is 76.
%e A125669 From _Andrew Howroyd_, May 25 2018: (Start)
%e A125669 Case n=7: illustrations of the 4 graphs:
%e A125669 o o o o o o o o---o o o---o
%e A125669 / \ / \ / \ / \ / / \ / \ / \ / \ \ \
%e A125669 o---o o---o o---o---o---o o---o---o---o o---o---o---o
%e A125669 (End)
%o A125669 (PARI) \\ here G is A000598 as series
%o A125669 G(n)={my(g=O(x)); for(n=1, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x^n)); g}
%o A125669 C1(n)={sum(i=1, n\2-1, sum(j=1, n\2-i, (d1^(2*(i+j)) + 2*d1^(2*i)*d2^j + d2^(i+j))*(1 + d1)^2))/(8*(1-d1))}
%o A125669 C2(n)={sum(k=1, n\4, 2*(d2^(2*k) + d4^k)*(1 + d2))*(1+d1)/(8*(1-d2))}
%o A125669 seq(n)={my(s=G(n)); my(d=x*(s^2+subst(s, x, x^2))/2); my(g(p,e)=subst(p + O(x*x^(n\e)), x, x^e)); Vec(O(x^n/x) + g(s,1)^2*substvec(C1(n-2),[d1,d2],[g(d,1),g(d,2)]) + g(s,2)*substvec(C2(n-2), [d1,d2,d4], [g(d,1),g(d,2),g(d,4)]))} \\ _Andrew Howroyd_, May 25 2018
%Y A125669 Cf. A121162, A121330, A121331, A121332, A121333, A125064, A125670.
%K A125669 nonn
%O A125669 6,2
%A A125669 _Parthasarathy Nambi_, Jan 29 2007
%E A125669 Terms a(16) and beyond from _Andrew Howroyd_, May 25 2018