A305132 - OEIS

%I #8 May 26 2018 16:28:07

%S 1,3,11,36,116,366,1151,3583,11093,34141,104489,318139,963899,2907276,

%T 8731919,26125538,77889504,231466147,685811867,2026481941,5973064855,

%U 17565416721,51547293439,150977445294,441409701444,1288409915625,3754926609800,10927779696264

%N Number of connected graphs on n unlabeled nodes with exactly 2 cycles joined along two or more edges but not more than half each cycle and all nodes having degree at most 4.

%C The resulting graph will actually have three cycles. See A121331 for the special case of all three cycles having the same length.

%C Equivalently, the number of connected simple graphs with n unlabeled nodes and n + 1 edges and all nodes having degree at most 4 (A112410) less those graphs described by A125669, A125670 and A125671.

%H Andrew Howroyd, <a href="/A305132/b305132.txt">Table of n, a(n) for n = 5..200</a>

%F a(n) >= A125672(n) + A125673(n).

%e Illustration of graphs for n=5 and n=6:

%e     o          o--o       o           o--o

%e    /|\        /|\        /|\         /|  |

%e   o o o      o o o      o o o--o    o o  |

%e    \|/        \|/        \|/         \|  |

%e     o          o          o           o--o

%o (PARI) \\ here G is A000598 as series

%o 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 C1(n)={subst(Pol(x^3*d1^3/(1-x*d1)^3 + 3*x^3*d1*d2/((1-x*d1)*(1-x^2*d2)) + 2*x^3*d3/(1-x^3*d3) + O(x*x^n)), x, 1)/12}

%o C2(n)={subst(Pol(((x*d1+x^2*d2)/(1-x^2*d2))^3 + 3*(x*d1+x^2*d2)*x^2*d2/(1-x^2*d2)^2 + 2*(x^3*d3 + x^6*d6)/(1-x^6*d6) + O(x*x^n)), x, 1)/12}

%o 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)]))}

%Y Cf. A112410, A121331, A125669, A125670, A125671, A125672, A125673.

%K nonn

%O 5,2

%A _Andrew Howroyd_, May 26 2018