cp's OEIS Frontend

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.

%I A125671 #14 May 25 2018 20:11:25
%S A125671 1,3,11,32,100,294,881,2590,7639,22344,65278,189832,550846,1593558,
%T A125671 4600435,13251623,38104280,109382300,313543725,897588156,2566575323,
%U A125671 7331196543,20921299025,59653124923,169959192844,483897197563,1376848221698,3915320424705,11128029239672
%N 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.
%C A125671 Here 'alpha' is the number of atoms the two rings have in common.
%C A125671 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
%H A125671 Andrew Howroyd, <a href="/A125671/b125671.txt">Table of n, a(n) for n = 4..200</a>
%H A125671 J. B. Hendrickson and C. A. Parks, <a href="https://doi.org/10.1021/ci00001a018">Generation and Enumeration of Carbon skeletons</a>, J. Chem. Inf. Comput. Sci., 31 (1991), pp. 101-107. See Table VII column 4 on page 104.
%e A125671 If n=5 then the number of bicyclics when 'alpha' = two is 3.
%e A125671 If n=6 then the number of bicyclics when 'alpha' = two is 11.
%e A125671 If n=7 then the number of bicyclics when 'alpha' = two is 32.
%e A125671 If n=8 then the number of bicyclics when 'alpha' = two is 100.
%e A125671 Case n=5: illustration of the 3 graphs:
%e A125671 .
%e A125671        o---o        o---o       o   o
%e A125671       /|   |       /|\         /|\  |
%e A125671      / |   |      / | \       / | \ |
%e A125671     /  |   |     /  |  \     /  |  \|
%e A125671    o---o---o    o---o---o   o---o---o
%o A125671 (PARI) \\ here G is A000598 as series
%o A125671 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}
%o A125671 C1(n)={(sum(k=2, n, (k-1)*d1^k) + sum(k=1, n\2, d2^k))/4}
%o A125671 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}
%o A125671 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
%Y A125671 Cf. A121165, A121330, A121331, A121332, A121333, A125064, A125670.
%K A125671 nonn
%O A125671 4,2
%A A125671 _Parthasarathy Nambi_, Jan 29 2007
%E A125671 a(4) prepended and terms a(16) and beyond from _Andrew Howroyd_, May 25 2018