A305132 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.

1, 3, 11, 36, 116, 366, 1151, 3583, 11093, 34141, 104489, 318139, 963899, 2907276, 8731919, 26125538, 77889504, 231466147, 685811867, 2026481941, 5973064855, 17565416721, 51547293439, 150977445294, 441409701444, 1288409915625, 3754926609800, 10927779696264

Offset

5,2

Comments

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

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.

Links

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

Formula

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

Example

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 o  |
   \|/        \|/        \|/         \|  |
    o          o          o           o--o

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)={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}
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}
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)]))}

Crossrefs

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

Sequence in context: A119126 A119088 A296391 * A291413 A347829 A017937

Adjacent sequences: A305129 A305130 A305131 * A305133 A305134 A305135

Keyword

nonn

Author

Andrew Howroyd, May 26 2018

Status

approved