A096202 -id:A096202 - 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

A096154 Number of tilings of {1...n} by translation and reflection of a single set.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 13, 6, 20, 2, 56, 2, 68, 12, 160, 2, 299, 2, 584, 18, 1028, 2, 2338, 8, 4100, 38, 8456, 2, 16576, 2, 33469, 30, 65540

Offset: 1

Jon Wild, Jul 26 2004

nonn

a(n) counts the partitions of {1...n} with the property that all elements of the partition are congruent, modulo translation and reflection, to the same tile.

Two tilings that are reflections of each other are considered distinct. E.g. {{1,2,6},{3,7,8},{4,5,9}} and {{1,5,6},{2,3,7},{4,8,9}} are both included in the count for a(9). The first tile that allows more than one tiling for the same set without one being a reflection of the other is {1,2,7} on the span {1...12}.

			a(8)=13 because the following are the 13 tilings of {1...8}:
{{1},{2},{3},{4},{5},{6},{7},{8}} tile: {1}
{{1,2},{3,4},{5,6},{7,8}} tile: {1,2}
{{1,3},{2,4},{5,7},{6,8}} tile: {1,3}
{{1,5},{2,6},{3,7},{4,8}} tile: {1,5}
{{1,2,3,4},{5,6,7,8}} tile: {1,2,3,4}
{{1,2,3,5},{4,6,7,8}} tile: {1,2,3,5}
{{1,5,6,7},{2,3,4,8}} tile: {1,2,3,7}
{{1,2,4,6},{3,5,7,8}} tile: {1,2,4,6}
{{1,4,6,7},{2,3,5,8}} tile: {1,2,4,7}
{{1,2,5,6},{3,4,7,8}} tile: {1,2,5,6}
{{1,3,4,7},{2,5,6,8}} tile: {1,3,4,7}
{{1,3,5,7},{2,4,6,8}} tile: {1,3,5,7}
{{1,2,3,4,5,6,7,8}} tile: {1,2,3,4,5,6,7,8}

Cf. A096202, A096203.

a(n)-4 often seems to be a power of 2. - Don Reble

More terms from Don Reble, Jul 04 2004

A329128 Number of nonequivalent sets whose translations and reflections cover {1..n}.

Original entry on oeis.org

1, 2, 3, 6, 8, 17, 24, 52, 77, 171, 265, 593, 952, 2131, 3519, 7846, 13238, 29351, 50374, 111031, 193155, 423403, 744616, 1624302, 2881784, 6260030, 11186219, 24213106, 43522800, 93922741, 169653109, 365172178

Offset: 1

Andrew Howroyd, Nov 07 2019

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

			For n = 4 there are 6 sets (up to equivalence) that with their reflections and translations cover {1..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}}.
.
For n = 5 there are 8 sets (up to equivalence) that with their reflections and translations cover {1..5}:
  {{1}, {2}, {3}, {4}, {5}};
  {{1, 2}, {2, 3}, {3, 4}, {4, 5}};
  {{1, 3}, {2, 4}, {3, 5}};
  {{1, 2, 4}, {1, 3, 4}, {2, 3, 5}, {2, 4, 5}};
  {{1, 2, 3}, {2, 3, 4}, {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. A079500 (if only translations allowed).

Cf. A096202, A096203, A329235.

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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A096154 Number of tilings of {1...n} by translation and reflection of a single set.

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A329128 Number of nonequivalent sets whose translations and reflections cover {1..n}.

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

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