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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.
4
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
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
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