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.

1, 2, 9, 26, 87, 257, 787, 2322, 6891, 20160, 58939, 171203, 496294, 1433558, 4132744, 11886827, 34133563, 97856500, 280172582, 801174478, 2288600128, 6531205571, 18622839635, 53059229091, 151067980960, 429840337630, 1222335365450, 3474107883033, 9869276762717

Offset: 5

Author

Parthasarathy Nambi, Jan 29 2007

Keywords

• nonn

Comments

Here 'alpha' is the number of atoms the two rings have in common.

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

Examples

			If n=5 then the number of bicyclics when 'alpha' = one is 1.
If n=6 then the number of bicyclics when 'alpha' = one is 2.
If n=7 then the number of bicyclics when 'alpha' = one is 9.
If n=8 then the number of bicyclics when 'alpha' = one is 26.
From _Andrew Howroyd_, May 24 2018: (Start)
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.
   o---o-----o     o---o---o
    \ / \    |      \ / \ /
     o   o---o       o   o---o
(End)
		

References

• 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.

Links

• Andrew Howroyd, Table of n, a(n) for n = 5..200

Crossrefs

Cf. A121158, A121330, A121331, A121332, A121333, A125064.

Programs

PARI

\\ here G is A000598 as series
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}
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}
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

Extensions

Terms a(16) and beyond from Andrew Howroyd, May 24 2018