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

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

Author

Jon Wild, Jul 26 2004

Keywords

• nonn

Comments

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}.

Examples

			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}
		

Crossrefs

Cf. A096202, A096203.

Formula

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

Extensions

More terms from Don Reble, Jul 04 2004