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

6,2

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

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.

Example

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)

Prog

(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

Crossrefs

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

Sequence in context: A196508 A121257 A145563 * A295116 A303508 A258627

Adjacent sequences: A125666 A125667 A125668 * A125670 A125671 A125672

Keyword

nonn

Author

Parthasarathy Nambi, Jan 29 2007

Extensions

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

Status

approved