login
A125670
Number of bicyclic skeletons with n carbon atoms and the parameter 'alpha' having the value of 1. See the paper by Hendrickson and Parks for details.
5
1, 2, 9, 26, 87, 257, 787, 2322, 6891, 20160, 58939, 171203, 496294, 1433558, 4132744, 11886827, 34133563, 97856500, 280172582, 801174478, 2288600128, 6531205571, 18622839635, 53059229091, 151067980960, 429840337630, 1222335365450, 3474107883033, 9869276762717
OFFSET
5,2
COMMENTS
Here 'alpha' is the number of atoms the two rings have in common.
Equivalently, the number of graphs on n unlabeled nodes with exactly 2 cycles joined at a single node and all nodes having degree at most 4. See A121158 for the special case of both cycles having the same length. - Andrew Howroyd, May 24 2018
REFERENCES
James B. Hendrickson and Camden A. Parks, "Generation and Enumeration of Carbon Skeletons", J. Chem. Inf. Comput. Sci., vol. 31 (1991), pp. 101-107. See Table VII column 3 on page 104.
LINKS
EXAMPLE
If n=5 then the number of bicyclics when 'alpha' = one is 1.
If n=6 then the number of bicyclics when 'alpha' = one is 2.
If n=7 then the number of bicyclics when 'alpha' = one is 9.
If n=8 then the number of bicyclics when 'alpha' = one is 26.
From Andrew Howroyd, May 24 2018: (Start)
Case n = 6: the two cases are a 3-cycle joined to a 4-cycle and a 3-cycle joined to another 3-cycle with a pendant edge.
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}
CycleIndex(n)={(sum(i=1, (n-1)\2-1, sum(j=1, (n-1)\2-i, (j1^(2*(i+j)) + 2*j1^(2*i)*j2^j + j2^(i+j))*(1 + j1)^2)) + sum(k=1, (n-1)\4, 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Parthasarathy Nambi, Jan 29 2007
EXTENSIONS
Terms a(16) and beyond from Andrew Howroyd, May 24 2018
STATUS
approved