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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 18 02:31 EDT 2021. Contains 347504 sequences. (Running on oeis4.)