A329128 -id:A329128 - cp's OEIS frontend

A096203 Number of coverings of {1..n} by translation and reflection of a single set.

1, 2, 3, 7, 18, 66, 239, 963, 3792, 15230, 60297, 240295, 952530, 3783000, 14999274, 59492918, 235852544, 935260075, 3707948564, 14702345112, 58294850481, 231152521791, 916584704599, 3634684693457, 14413639272087

Offset: 1

Jon Wild, Jul 27 2004

nonn

			a(4) = 7 because the following are the 7 coverings of {1...4}, each one of which only uses a single set and its translations and reflections:
  {{1}, {2}, {3}, {4}};
  {{1, 2}, {3, 4}};
  {{1, 2}, {2, 3}, {3, 4}};
  {{1, 3}, {2, 4}};
  {{1, 2, 4}, {1, 3, 4}};
  {{1, 2, 3}, {2, 3, 4}};
  {{1, 2, 3, 4}}.
.
a(5) = 18 because the following are the 18 coverings of {1...5}, each one of which only uses a single set and its translations and reflections:
  {{1}, {2}, {3}, {4}, {5}};
  {{1, 2}, {2, 3}, {3, 4}, {4, 5}};
  {{1, 2}, {2, 3}, {4, 5}};
  {{1, 2}, {3, 4}, {4, 5}};
  {{1, 3}, {2, 4}, {3, 5}};
  {{1, 2, 4}, {1, 3, 4}, {2, 3, 5}, {2, 4, 5}};
  {{1, 2, 4}, {1, 3, 4}, {2, 3, 5}};
  {{1, 2, 4}, {1, 3, 4}, {2, 4, 5}};
  {{1, 2, 4}, {2, 3, 5}, {2, 4, 5}};
  {{1, 3, 4}, {2, 3, 5}, {2, 4, 5}};
  {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}};
  {{1, 2, 4}, {2, 3, 5}};
  {{1, 3, 4}, {2, 3, 5}};
  {{1, 3, 4}, {2, 4, 5}};
  {{1, 2, 3}, {3, 4, 5}};
  {{1, 2, 3, 5}, {1, 3, 4, 5}};
  {{1, 2, 3, 4}, {2, 3, 4, 5}};
  {{1, 2, 3, 4, 5}}.

Cf. A096202 (if only translations allowed).

Cf. A096154, A329128.

Corrected by Andrew Howroyd, Nov 08 2019

A329235 Number of nonequivalent symmetric sets whose translations cover {1..n}.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 12, 19, 23, 36, 44, 68, 84, 128, 161, 243, 308, 462, 592, 882, 1140, 1690, 2200, 3249, 4255, 6264, 8246, 12110, 16008, 23466, 31128, 45566, 60618, 88644, 118205, 172731, 230782, 337072, 451082, 658628, 882582, 1288432, 1728484, 2523104, 3388084

Offset: 1

Andrew Howroyd, Nov 08 2019

nonn

Equivalence is up to translation. Only translations that are subsets of {1..n} are included.

Symmetric sets are those such that the set remains unchanged after mapping each element x to m - x, where m is the sum of the greatest and least elements. All sets of at most two elements are symmetric.

			For n = 6 there are 10 symmetric sets (up to equivalence) that with their translations cover {1..6}:
  {{1}, {2}, {3}, {4}, {5}, {6}};
  {{1, 4}, {2, 5}, {3, 6}};
  {{1, 3}, {2, 4}, {3, 5}, {4, 6}};
  {{1, 3, 5}, {2, 4, 6}};
  {{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}};
  {{1, 2, 4, 5}, {2, 3, 5, 6}};
  {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}, {4, 5, 6}};
  {{1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}};
  {{1, 2, 3, 4, 5}, {2, 3, 4, 5, 6}};
  {{1, 2, 3, 4, 5, 6}}.

Cf. A079500 (if symmetry is not required).

Cf. A096202, A329128.

cp's OEIS Frontend

A096203 Number of coverings of {1..n} by translation and reflection of a single set.

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