OFFSET
1,55
COMMENTS
A half-arc-transitive graph is vertex-transitive and edge-transitive but not arc-transitive.
The terms a(1)-a(1000) are complete.
The 339 nonzero terms among a(1), ..., a(1000) sum to 3246.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1000
Primož Potočnik, Pablo Spiga and Gabriel Verret, A census of 4-valent half-arc-transitive graphs and arc-transitive digraphs of valence two.
Primož Potočnik, Pablo Spiga and Gabriel Verret, Census-HAT-1k.txt.
Eric Weisstein's World of Mathematics, Doyle Graph.
Eric Weisstein's World of Mathematics, Half-Arc-Transitive Graph.
Eric Weisstein's World of Mathematics, Quartic Graph.
EXAMPLE
a(n) = 0 for 1 <= n <= 26.
a(27) = 1 since the Doyle graph is the unique connected 4-valent half-arc-transitive graph on 27 vertices.
a(55) = 2 since there are two connected 4-valent half-arc-transitive graphs on 55 vertices.
MATHEMATICA
With[{orders = ToExpression /@ StringCases[Import["https://users.fmf.uni-lj.si/potocnik/ATDcensus-OnLine/Census-HAT-1k.txt", "Text"], StartOfLine ~~ (n : DigitCharacter ..) ~~ "|" ~~ DigitCharacter .. ~~ EndOfLine :> n]}, Lookup[Counts[orders], Range[1000], 0]]
CROSSREFS
Cf. A006820 (number of connected quartic graphs on n vertices).
Cf. A059282 (number of connected cubic arc-transitive graphs on 2n vertices).
Cf. A133181 (number of connected vertex- and edge-transitive graphs on n vertices).
Cf. A286280 (number of connected arc-transitive graphs on n vertices).
Cf. A398352 (orders n for which a(n) > 0).
KEYWORD
nonn,hard,more
AUTHOR
Eric W. Weisstein, Jul 26 2026
STATUS
approved