A121158 -id:A121158 - cp's OEIS frontend

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

N. J. A. Sloane, Aug 27 2006

nonn

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

			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)

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).

Cf. A121158, A121162, A121165, A305132.

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 *)

\\ 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

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

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

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.

Original entry on oeis.org

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

Parthasarathy Nambi, Jan 29 2007

nonn

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

			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)

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.

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

Cf. A121158, A121330, A121331, A121332, A121333, A125064.

\\ 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

Terms a(16) and beyond from Andrew Howroyd, May 24 2018

A121162 Number of separated bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

Original entry on oeis.org

1, 3, 13, 41, 141, 440, 1391, 4244, 12913, 38651, 115082, 339646, 997709, 2915010, 8485573, 24612666, 71191458, 205393819, 591330506, 1699226719, 4874925420, 13965498369, 39957144189, 114193222891, 326023307022, 929958622555, 2650483647976, 7548608038736

Offset: 6

Parthasarathy Nambi, Aug 13 2006

nonn

Equivalently, the number of connected graphs on n unlabeled nodes with exactly 2 cycles of equal length without any shared node and all nodes having degree at most 4. - Andrew Howroyd, May 25 2018

Andrew Howroyd, Table of n, a(n) for n = 6..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).

Cf. A121158, A125669.

\\ 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\4, d1^(4*k) + 2*d1^(2*k)*d2^k + d2^(2*k))*(1 + d1^2)/(8*(1-d1))}
C2(n)={sum(k=1, n\4,  2*(d2^(2*k) + d4^k)*(1 + d2))*(1+d1)/(8*(1-d2))}
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

More terms from N. J. A. Sloane, Aug 27 2006

Terms a(26) and beyond from Andrew Howroyd, May 25 2018

A107278 Number of spiro bicyclic skeletons with n carbon atoms.

Original entry on oeis.org

1, 4, 17, 60, 209, 685, 2204, 6913, 21387, 65241, 197104, 590284, 1755804, 5191723, 15276610, 44761350, 130682371, 380337474, 1103927291, 3196494453

Offset: 6

N. J. A. Sloane, Sep 10 2006

nonn

See Parks et al. for precise definition. Sequence is last column of table III.

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

Table III of the Parks et al. reference has 4 sequences: A121158, A121159, A121160 and this sequence.

Incorrect g.f. removed by Georg Fischer, May 24 2019

cp's OEIS Frontend

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

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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.

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A121162 Number of separated bicyclic skeletons with n carbon atoms (see Parks et al. for precise definition).

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A107278 Number of spiro bicyclic skeletons with n carbon atoms.

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