A121158 Number of spiro bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

1, 1, 5, 10, 34, 82, 240, 630, 1764, 4752, 13108, 35703, 98108, 268553, 737864, 2025779, 5572160, 15331017, 42230755, 116395955, 321089489, 886320404, 2448312482, 6767186801, 18716207007, 51792971141, 143403624284, 397254931272, 1101003729796, 3052855074597

Offset: 5

Author

Parthasarathy Nambi, Aug 13 2006

Keywords

• nonn

Comments

Equivalently, the number of graphs on n unlabeled nodes with exactly 2 cycles of equal length joined at a single node and all nodes having degree at most 4. - Andrew Howroyd, May 24 2018

References

• Camden A. Parks and James B. Hendrickson, Enumeration of monocyclic and bicyclic carbon skeletons, J. Chem. Inf. Comput. Sci., vol. 31, 334-339 (1991).

Links

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

Crossrefs

Cf. A107278, A121158, A121159, A121160.

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(k=1, (n-1)\4,  (j1^(4*k) + 2*j1^(2*k)*j2^k + j2^(2*k))*(1 + j1^2) + 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(26) and beyond from Andrew Howroyd, May 24 2018