A243243 Number of unlabeled, connected graphs on n vertices with at least one subgraph isomorphic to a C_4, where C_4 is the cycle graph on four vertices.

0, 0, 0, 3, 13, 93, 796, 10931, 260340, 11713182, 1006682063, 164059710255, 50335906936959, 29003487454251217, 31397381142667479256, 63969560113223974443840, 245871831682084008526845525, 1787331725248899088577102145274, 24636021429399867655316345340289103

Offset: 1

Author

Travis Hoppe and Anna Petrone, Jun 01 2014

Keywords

• nonn

Links

• Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
• T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
• Eric Weisstein's World of Mathematics, Square-Free Graph

Crossrefs

Cf. A001349, A077269.

Programs

Mathematica

terms = 19;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a88[n_] := Module[{s = 0}, Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!];
A001349 = EULERi[Array[a88, terms]];
A006786 = {1, 2, 4, 8, 18, 44, 117, 351, 1230, 5069, 25181, 152045, 1116403, 9899865, 104980369, 1318017549, 19427531763, 333964672216, 6660282066936};
A077269 = EULERi[A006786];
A001349 - A077269 (* Jean-François Alcover, Feb 15 2019, after Andrew Howroyd in A001349 and A077269 *)

Formula

a(n) = A001349(n) - A077269(n).

Extensions

a(11)-a(17) using formula from Falk Hüffner, Jan 15 2016

a(18)-a(19) from Jean-François Alcover, Feb 15 2019 using Andrew Howroyd's code.