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A125670 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 1. See the paper by Hendrickson and Parks for details.

%I A125670 #8 May 24 2018 22:13:58
%S A125670 1,2,9,26,87,257,787,2322,6891,20160,58939,171203,496294,1433558,
%T A125670 4132744,11886827,34133563,97856500,280172582,801174478,2288600128,
%U A125670 6531205571,18622839635,53059229091,151067980960,429840337630,1222335365450,3474107883033,9869276762717
%N A125670 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 1. See the paper by Hendrickson and Parks for details.
%C A125670 Here 'alpha' is the number of atoms the two rings have in common.
%C A125670 Equivalently, the number of graphs on n unlabeled nodes with exactly 2 cycles joined at a single node and all nodes having degree at most 4. See A121158 for the special case of both cycles having the same length. - _Andrew Howroyd_, May 24 2018
%D A125670 James B. Hendrickson and Camden A. Parks, "Generation and Enumeration of Carbon Skeletons", J. Chem. Inf. Comput. Sci., vol. 31 (1991), pp. 101-107. See Table VII column 3 on page 104.
%H A125670 Andrew Howroyd, <a href="/A125670/b125670.txt">Table of n, a(n) for n = 5..200</a>
%e A125670 If n=5 then the number of bicyclics when 'alpha' = one is 1.
%e A125670 If n=6 then the number of bicyclics when 'alpha' = one is 2.
%e A125670 If n=7 then the number of bicyclics when 'alpha' = one is 9.
%e A125670 If n=8 then the number of bicyclics when 'alpha' = one is 26.
%e A125670 From _Andrew Howroyd_, May 24 2018: (Start)
%e A125670 Case n = 6: the two cases are a 3-cycle joined to a 4-cycle and a 3-cycle joined to another 3-cycle with a pendant edge.
%e A125670    o---o-----o     o---o---o
%e A125670     \ / \    |      \ / \ /
%e A125670      o   o---o       o   o---o
%e A125670 (End)
%o A125670 (PARI) \\ here G is A000598 as series
%o A125670 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 A125670 CycleIndex(n)={(sum(i=1, (n-1)\2-1, sum(j=1, (n-1)\2-i, (j1^(2*(i+j)) + 2*j1^(2*i)*j2^j + j2^(i+j))*(1 + j1)^2)) + sum(k=1, (n-1)\4,  2*(j2^(2*k) + j4^k)*(1 + j2)))/8}
%o A125670 seq(n)={my(t=G(n)); t=x*(t^2+subst(t, x, x^2))/2; my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x^n)); Vec(substvec(CycleIndex(n), [j1,j2,j4], [g(1),g(2),g(4)]))} \\ _Andrew Howroyd_, May 24 2018
%Y A125670 Cf. A121158, A121330, A121331, A121332, A121333, A125064.
%K A125670 nonn
%O A125670 5,2
%A A125670 _Parthasarathy Nambi_, Jan 29 2007
%E A125670 Terms a(16) and beyond from _Andrew Howroyd_, May 24 2018