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 A121331 Number of bridged bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition). 8
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified August 5 02:16 EDT 2024. Contains 374935 sequences. (Running on oeis4.)