A372174 Number of unlabeled simple graphs covering n vertices with a unique triangle.
0, 0, 0, 1, 1, 5, 16, 79, 424, 3098, 28616
Offset: 0
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First differences of A372194.
This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The a(0) = 1 through a(5) = 10 simple graphs:
{} . {12} {12-13} {12-34} {12-13-45}
{12-13-23} {12-13-14} {12-13-14-15}
{12-13-24} {12-13-14-25}
{12-13-14-23} {12-13-23-45}
{12-13-24-34} {12-13-24-35}
{12-13-14-15-23}
{12-13-14-23-25}
{12-13-14-23-45}
{12-13-14-25-35}
{12-13-24-35-45}
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}],{0,n}], Union@@#==Range[n]&]]],{n,0,5}]
\\ G defined in A008406. a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024
The a(1) = 1 through a(4) = 9 graph edge sets:
{} {} {} {}
{12} {12} {12}
{12-13} {12-13}
{12-13-23} {12-34}
{12-13-14}
{12-13-23}
{12-13-24}
{12-13-14-23}
{12-13-24-34}
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]], Length[#]<=n&]]],{n,0,5}]
a(n) = if(n<=1, n>=0, polcoef(G(n, O(x*x^n))/(1-x),n)) \\ G(n) defined in A008406. - Andrew Howroyd, Feb 20 2024
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