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A372194 Number of unlabeled graphs with n vertices and a unique triangle.

%I A372194 #16 Aug 03 2024 14:35:55
%S A372194 0,0,0,1,2,7,23,102,526,3624,32240,382095,5986945
%N A372194 Number of unlabeled graphs with n vertices and a unique triangle.
%C A372194 The labeled version is A372172.
%H A372194 Gus Wiseman, <a href="/A372194/a372194.png">The a(3) = 1 through a(6) = 23 graphs with a unique triangle</a>.
%F A372194 First differences are A372174.
%e A372194 Representatives of the a(3) = 1 through a(6) = 23 graphs:
%e A372194     12,13,23    12,13,23       12,13,23             12,13,23
%e A372194                 14,23,24,34    12,34,35,45          12,34,35,45
%e A372194                                14,23,24,34          14,23,24,34
%e A372194                                12,25,34,35,45       12,25,34,35,45
%e A372194                                14,25,34,35,45       12,36,45,46,56
%e A372194                                15,25,34,35,45       13,23,45,46,56
%e A372194                                12,14,25,34,35,45    14,25,34,35,45
%e A372194                                                     15,25,34,35,45
%e A372194                                                     12,14,25,34,35,45
%e A372194                                                     12,23,36,45,46,56
%e A372194                                                     13,23,36,45,46,56
%e A372194                                                     13,25,36,45,46,56
%e A372194                                                     13,26,36,45,46,56
%e A372194                                                     14,25,36,45,46,56
%e A372194                                                     15,26,36,45,46,56
%e A372194                                                     16,26,36,45,46,56
%e A372194                                                     12,13,25,36,45,46,56
%e A372194                                                     12,13,26,36,45,46,56
%e A372194                                                     13,23,25,36,45,46,56
%e A372194                                                     14,23,25,36,45,46,56
%e A372194                                                     16,23,25,36,45,46,56
%e A372194                                                     13,14,23,25,36,45,46,56
%e A372194                                                     13,15,23,25,36,45,46,56
%o A372194 (nauty) geng $n | countg -T1 # _Georg Grasegger_, Aug 03 2024
%Y A372194 For no triangles we have A006785, covering A372169.
%Y A372194 Column k = 1 of A263340, covering A372173.
%Y A372194 The labeled version is A372172.
%Y A372194 The covering case is A372174, labeled A372171.
%Y A372194 For all cycles (not just triangles): A236570, A372193, A372191, A372195.
%Y A372194 A000088 counts unlabeled graphs, labeled A006125.
%Y A372194 A001858 counts acyclic graphs, unlabeled A005195.
%Y A372194 A002494 counts unlabeled covering graphs, labeled A006129.
%Y A372194 A372176 counts labeled graphs by directed cycles, covering A372175.
%Y A372194 Cf. A000055, A137917, A137918, A140637, A144958, A213434, A372168, A372170.
%K A372194 nonn,more
%O A372194 0,5
%A A372194 _Gus Wiseman_, Apr 24 2024
%E A372194 a(11)-a(12) added by _Georg Grasegger_, Aug 03 2024