A125671 Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 2. See the paper by Hendrickson and Parks for details.

1, 3, 11, 32, 100, 294, 881, 2590, 7639, 22344, 65278, 189832, 550846, 1593558, 4600435, 13251623, 38104280, 109382300, 313543725, 897588156, 2566575323, 7331196543, 20921299025, 59653124923, 169959192844, 483897197563, 1376848221698, 3915320424705, 11128029239672

Offset

4,2

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 joined at a single edge and all nodes having degree at most 4. See A121165 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 = 4..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 4 on page 104.

Example

If n=5 then the number of bicyclics when 'alpha' = two is 3.
If n=6 then the number of bicyclics when 'alpha' = two is 11.
If n=7 then the number of bicyclics when 'alpha' = two is 32.
If n=8 then the number of bicyclics when 'alpha' = two is 100.
Case n=5: illustration of the 3 graphs:
.
       o---o        o---o       o   o
      /|   |       /|\         /|\  |
     / |   |      / | \       / | \ |
    /  |   |     /  |  \     /  |  \|
   o---o---o    o---o---o   o---o---o

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(k=2, n, (k-1)*d1^k) + sum(k=1, n\2, d2^k))/4}
C2(n)={(sum(k=1, n\2, d2^k) + sum(i=1, n-1, sum(j=1, n-i, d2^(i\2+j\2) * d1^(i%2+j%2))))/4}
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. A121165, A121330, A121331, A121332, A121333, A125064, A125670.

Sequence in context: A353065 A321645 A296315 * A034561 A131747 A295626

Adjacent sequences: A125668 A125669 A125670 * A125672 A125673 A125674

Keyword

nonn

Author

Parthasarathy Nambi, Jan 29 2007

Extensions

a(4) prepended and terms a(16) and beyond from Andrew Howroyd, May 25 2018

Status

approved