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.

1, 4, 20, 76, 288, 1005, 3433, 11324, 36712, 116809, 367076, 1140226, 3510491, 10722708, 32539364, 98178211, 294767639, 881147521, 2623934079, 7787024985, 23039064263, 67977412951, 200072442611, 587532484513, 1721812143140, 5036454320102, 14706743476128

Offset: 6

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 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

Examples

			If n=6 then the number of bicyclics when 'alpha' = zero is 1.
If n=7 then the number of bicyclics when 'alpha' = zero is 4.
If n=8 then the number of bicyclics when 'alpha' = zero is 20.
If n=9 then the number of bicyclics when 'alpha' = zero is 76.
From _Andrew Howroyd_, May 25 2018: (Start)
Case n=7: illustrations of the 4 graphs:
     o   o   o       o   o   o       o   o---o       o   o---o
    / \ / \ / \     / \ /   / \     / \     / \     / \   \   \
   o---o   o---o   o---o---o---o   o---o---o---o   o---o---o---o
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 6..200
• J. B. Hendrickson and C. A. Parks, Generation and Enumeration of Carbon skeletons, J. Chem. Inf. Comput. Sci., 31 (1991), pp. 101-107. See Table VII column 2 on page 104.

Crossrefs

Cf. A121162, A121330, A121331, A121332, A121333, A125064, A125670.

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}
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))}
C2(n)={sum(k=1, n\4,  2*(d2^(2*k) + d4^k)*(1 + d2))*(1+d1)/(8*(1-d2))}
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

Extensions

Terms a(16) and beyond from Andrew Howroyd, May 25 2018