A121331 Number of bridged bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

1, 2, 6, 15, 39, 99, 258, 671, 1762, 4657, 12372, 33036, 88590, 238483, 644045, 1744542, 4737341, 12894158, 35165994, 96083192, 262951511, 720685274, 1977846334, 5434588909, 14949284828, 41163690109, 113451949753, 312955174089, 863965424349, 2386874582238

Offset

5,2

Comments

Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles of the same even length joined along half their length and all nodes having degree at most 4. The resulting graph will have three equal length cycles. - Andrew Howroyd, May 25 2018

Links

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

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

Formula

a(n) ~ c * d^n / sqrt(n), where d = 1/A261340 = 2.815460033176150746526616778..., c = 0.0064202170754... . - Vaclav Kotesovec, Sep 08 2019

Example

From Andrew Howroyd, May 25 2018: (Start)
Illustration of graphs for n=5 and n=6:
    o          o--o       o
   /|\        /|\        /|\
  o o o      o o o      o o o--o
   \|/        \|/        \|/
    o          o          o
.
Illustration of graphs for n=7:
    o--o      o--o--o   o--o        o        o          o   o
   /|\       /|\       /|\         /|\      /|\        /|\ /
  o o o     o o o     o o o--o    o o o    o o o--o   o o o
   \|/       \|/       \|/       / \|/ \    \|/   |    \|/ \
    o--o      o         o       o   o   o    o    o     o   o
(End)

Mathematica

G[n_] := Module[{g}, g[_] = 0; Do[g[x_] = 1 + x*(g[x]^3/6 + g[x^2]*g[x]/2 + g[x^3]/3) + O[x]^n // Normal, {n}]; g[x]];
C1[n_] := Sum[(d1^(3*k)+3*d1^k*d2^k + 2*d3^k), {k, 1, Quotient[n, 3]}]/12;
C2[n_] := Sum[(d1^Mod[k, 2]*d2^Quotient[k, 2])^3 + 3*d1^Mod[k, 2]* d2^(Quotient[k, 2] + k) + 2*d3^Mod[k, 2]*d6^Quotient[k, 2], {k, 1, Quotient[n, 3]}]/12;
seq[n_] := Module[{s, d, g}, s = G[n]; d = x*(s^2 + (s /. x -> x^2))/2; g[p_, e_] := Normal[(p+O[x]^(Quotient[n, e]+1))] /. x :> x^e; g[s, 1]^2* (C1[n-2] /. Thread[{d1, d2, d3} :> {g[d, 1], g[d, 2], g[d, 3]}]) + g[s, 2]*(C2[n-2] /. Thread[{d1, d2, d3, d6} :> {g[d, 1], g[d, 2], g[d, 3], g[d, 6]}]) + O[x]^n] // CoefficientList[#, x]& // Drop[#, 3]&;
seq[33] (* Jean-François Alcover, Sep 08 2019, after Andrew Howroyd *)

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=1, n\3, (d1^(3*k) + 3*d1^k*d2^k + 2*d3^k))/12}
C2(n)={sum(k=1, n\3, (d1^(k%2)*d2^(k\2))^3 + 3*d1^(k%2)*d2^(k\2+k) + 2*d3^(k%2)*d6^(k\2))/12}
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, d3], [g(d, 1), g(d, 2), g(d, 3)]) + g(s, 2)*substvec(C2(n-2), [d1, d2, d3, d6], [g(d, 1), g(d, 2), g(d, 3), g(d, 6)]))} \\ Andrew Howroyd, May 25 2018

Crossrefs

Cf. A121158, A121162, A121165, A305132.

Sequence in context: A101522 A094969 A001674 * A026270 A321646 A246563

Adjacent sequences: A121328 A121329 A121330 * A121332 A121333 A121334

Keyword

nonn

Author

N. J. A. Sloane, Aug 27 2006

Extensions

Corrected by Franklin T. Adams-Watters and T. D. Noe, Oct 25 2006

a(24) corrected and terms a(26) and beyond from Andrew Howroyd, May 25 2018

Status

approved