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Number of nonequivalent sets whose translations and reflections cover {1..n}.
2

%I #18 Nov 08 2019 22:27:24

%S 1,2,3,6,8,17,24,52,77,171,265,593,952,2131,3519,7846,13238,29351,

%T 50374,111031,193155,423403,744616,1624302,2881784,6260030,11186219,

%U 24213106,43522800,93922741,169653109,365172178

%N Number of nonequivalent sets whose translations and reflections cover {1..n}.

%C Equivalence is up to translation and reflection. Only translations and reflections that are subsets of {1..n} are included.

%e For n = 4 there are 6 sets (up to equivalence) that with their reflections and translations cover {1..4}:

%e {{1}, {2}, {3}, {4}};

%e {{1, 2}, {2, 3}, {3, 4}};

%e {{1, 3}, {2, 4}};

%e {{1, 2, 4}, {1, 3, 4}};

%e {{1, 2, 3}, {2, 3, 4}};

%e {{1, 2, 3, 4}}.

%e .

%e For n = 5 there are 8 sets (up to equivalence) that with their reflections and translations cover {1..5}:

%e {{1}, {2}, {3}, {4}, {5}};

%e {{1, 2}, {2, 3}, {3, 4}, {4, 5}};

%e {{1, 3}, {2, 4}, {3, 5}};

%e {{1, 2, 4}, {1, 3, 4}, {2, 3, 5}, {2, 4, 5}};

%e {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}};

%e {{1, 2, 3, 5}, {1, 3, 4, 5}};

%e {{1, 2, 3, 4}, {2, 3, 4, 5}};

%e {{1, 2, 3, 4, 5}}.

%Y Cf. A079500 (if only translations allowed).

%Y Cf. A096202, A096203, A329235.

%K nonn,more

%O 1,2

%A _Andrew Howroyd_, Nov 07 2019