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

0,8

Links

Table of n, a(n) for n=0..23.

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

Formula

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

Example

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.

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 *)

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).

Sequence in context: A009225 A157612 A348103 * A286946 A184596 A149978

Adjacent sequences: A184940 A184941 A184942 * A184944 A184945 A184946

Keyword

nonn,hard,more

Author

Jason Kimberley, Jan 25 2011

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

Status

approved