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%I #25 Mar 17 2020 12:12:23
%S 0,0,0,0,0,1,1,2,5,16,57,263,1532,10747,87948,803885,8020590,86027734,
%T 983417704,11913817317,152352034707,2050055948375,28951137255862,
%U 428085461764471
%N Number of connected 4-regular simple graphs on n vertices with girth exactly 3.
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/C_k-reg_girth_eq_g_index">Index of sequences counting connected k-regular simple graphs with girth exactly g</a>
%F a(n) = A006820(n) - A033886(n).
%e 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.
%e The a(5)=1 complete graph on 5 vertices is 4-regular; it has 10 edges and 10 triangles.
%t A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];
%t A006820 = A@006820; A033886 = A@033886;
%t a[n_] := A006820[[n + 1]] - A033886[[n + 1]];
%t a /@ Range[0, 22] (* _Jean-François Alcover_, Jan 27 2020 *)
%Y 4-regular simple graphs with girth exactly 3: this sequence (connected), A185043 (disconnected), A185143 (not necessarily connected).
%Y 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).
%Y Connected 4-regular simple graphs with girth at least g: A006820 (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
%Y Connected 4-regular simple graphs with girth exactly g: this sequence (g=3), A184944 (g=4), A184945 (g=5).
%K nonn,hard,more
%O 0,8
%A _Jason Kimberley_, Jan 25 2011
%E 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
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