A184943 Number of connected 4-regular simple graphs on n vertices with girth exactly 3.

0, 0, 0, 0, 0, 1, 1, 2, 5, 16, 57, 263, 1532, 10747, 87948, 803885, 8020590, 86027734, 983417704, 11913817317, 152352034707, 2050055948375, 28951137255862, 428085461764471

Offset: 0

Author

Jason Kimberley, Jan 25 2011

Keywords

• nonn
• hard
• more

Examples

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 4-regular and connected, since it is acyclic, it has infinite girth.
The a(5)=1 complete graph on 5 vertices is 4-regular; it has 10 edges and 10 triangles.
		

Links

• Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Crossrefs

4-regular simple graphs with girth exactly 3: this sequence (connected), A185043 (disconnected), A185143 (not necessarily connected).

Connected k-regular simple graphs with girth exactly 3: A006923 (k=3), this sequence (k=4), A184953 (k=5), A184963 (k=6), A184973 (k=7), A184983 (k=8), A184993 (k=9).

Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).

Connected 4-regular simple graphs with girth exactly g: this sequence (g=3), A184944 (g=4), A184945 (g=5).

Programs

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A006820 = A@006820; A033886 = A@033886;
a[n_] := A006820[[n + 1]] - A033886[[n + 1]];
a /@ Range[0, 22] (* Jean-François Alcover, Jan 27 2020 *)

Formula

a(n) = A006820(n) - A033886(n).

Extensions

Term a(22) corrected and a(23) appended, due to the correction and extension of A006820 by Andrew Howroyd, from Jason Kimberley, Mar 13 2020